Quick recall · 319 cards
Short MCQ-style retrieval prompts. Tap a card to reveal the answer.
PYQ
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The average of the first 125 natural numbers is:
B · 63
PYQ
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If one-third of one-fourth of a number is 15, then three-tenths of that number is:
C · 45
PYQ
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Which of the following are whole numbers? −3, 0, 2.5, 4, 5, −1.2
C · 0, 4, 5
PYQ
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Two consecutive even numbers, when squared and added together, give a total of 244. What are these two numbers?
B · B) 12, 14
PYQ
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Which of the following is a prime number? (From context: options include numbers like 2,3,5,7,11,13,17,...)
C · C) 17
Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. 1 is not prime. 15=3×5, 25=5×5 composite. 17 is prime (divisors only 1,17). Hence C.[1]
PYQ
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Which of the following is a prime number?
C · 37
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13 + 23 + 33 + ... + 63 = ?
B · B. 441
The sum of cubes from 1^3 to n^3 is given by formula \( \left( \frac{n(n+1)}{2} \right)^2 \). Here n=6, sum = \( \left( \frac{6 \times 7}{2} \right)^2 = (21)^2 = 441 \). Option B matches.
PYQ
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Sum of digits of a two-digit number equals 9. Furthermore, the difference between these digits is 3. What is the number?
A · A) 63
Let the number be 10x + y, where x is tens digit, y is units digit. Given: x + y = 9, x - y = 3. Adding equations: 2x = 12 => x = 6. Then y = 3. Number is 63. Option A is correct.
PYQ
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Determine whether the given numbers are rational or irrational: (a) 1.33 (b) 0.1 (c) 0 (d) √5
A · (a) Rational, (b) Rational, (c) Rational, (d) Irrational
PYQ
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____ is the identity for the addition of rational numbers.
B · (b) 0
PYQ · 2020
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Which of the following statements is NOT always true?
A · The product of two irrational numbers is irrational
PYQ
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Determine if 3/5 is rational or irrational.
A · Rational
PYQ
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Which expression represents an irrational number?
C · √36 - 7
PYQ · 2024
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The LCM of two prime numbers p and q (p > q) is 221. Then the value of 3p – q is:
(a) 4
(b) 28
(c) 38
(d) 48
C · 38
LCM of two primes p and q (p > q) is pq = 221. Factorize 221 = 13 × 17. So p = 17, q = 13. Then 3p - q = 3(17) - 13 = 51 - 13 = 38. Option (c) matches this value[1].
PYQ · 2024
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A pair of irrational numbers whose product is a rational number is
(a) (√16, √4)
(b) (√5, √2)
(c) (√3, √27)
(d) (√36, √2)
C · (√3, √27)
PYQ · 2024
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The smallest irrational number by which √20 should be multiplied so as to get a rational number, is:
(a) √20
(b) √2
(c) 5
(d) √5
D · √5
PYQ
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The mean of the first 7 odd prime numbers is:
A · 12
PYQ
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How many prime numbers are there between 6 and 42?
C · 10
Prime numbers between 6 and 42 are: 7, 11, 13, 17, 19, 23, 29, 31, 37, 41. Total count = 10. Thus, correct option is C[2].
PYQ
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The product of the first three prime numbers is 1771. If x, y, z, w are four prime numbers in ascending order, what is the product of the last two prime numbers (z × w)?
A · 1357
Given xyz = 1771. Factorizing: 7 × 11 × 23 = 1771, so x=7, y=11, z=23. Given z + w = 82, so 23 + w = 82, w = 59 (prime). Thus, z × w = 23 × 59 = 1357. Option A[2].
PYQ
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There are four prime numbers written in ascending order of magnitude. The product of the first three is 7429 and the product of the last three is 12673. Find the first number.
B · 17
PYQ
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Which of the following are prime numbers? (i) 247 (ii) 397 (iii) 423
B · Only (ii)
(i) 247 = 13×19, not prime. (ii) 397 is prime (no divisors up to √397≈19.9: not divisible by 2,3,5,7,11,13,17,19). (iii) 423=3×141, not prime. Only (ii) is prime. Option B[6].
PYQ
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Which of these numbers is composite?
A. 3
B. 11
C. 12
D. 19
C · 12
PYQ
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How many factors of the number \( 2^8 \times 3^6 \times 5^4 \times 10^5 \) are multiples of 120?
C · 594
PYQ
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If both \( 11^2 \) and \( 3^3 \) are factors of the number \( a \times 4^3 \times 6^2 \times 13^{11} \), then what is the smallest possible value of 'a'?
A · 121
PYQ
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What is the prime factorization of 16 as a product of prime factors?
B · 2⁴
PYQ
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What is the GCD of 12 and 15?
B · 3
Prime factorization: 12 = \(2^2 \times 3\), 15 = \(3 \times 5\).Common factor: 3.Euclidean algorithm: GCD(15,12) = GCD(12,15 mod 12)=GCD(12,3)=3.Thus, GCD is 3, which corresponds to option B.[4]
PYQ
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GCD of 17 and 19
A · 1
17 and 19 are consecutive prime numbers, so they have no common factors other than 1.Euclidean: GCD(19,17)=GCD(17,2)=GCD(2,1)=1.Thus, GCD=1, option A.[4]
PYQ · 2018
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HCF of 2472, 1284 and a third number ‘n’ is 12. If their LCM is 8*9*5*10^3*107, then the number ‘n’ is:
A · 2^2*3^2*5^1
PYQ · 2021
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The HCF and LCM of two numbers are 11 and 385 respectively. If one number lies between 75 and 125, then that number is?
B · 88
PYQ · 2019
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If HCF of 189 and 297 is 27, find their LCM.
A · 2079
PYQ
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Which number is divisible by 2? 75, 45, 46, 49
C · C. 46
PYQ
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Which number is divisible by 4? 34, 51, 68, 38
C · C. 68
PYQ
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Is 1440 divisible by 15?
A · A. Yes
A number is divisible by 15 if it is divisible by both 3 and 5. For 1440: Ends with 0, so divisible by 5. Sum of digits: 1+4+4+0=9, divisible by 3. Hence, 1440 is divisible by 15. Option A.
PYQ
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Is 99992 divisible by 8?
A · A. Yes
Divisibility by 8: last three digits (992) must be divisible by 8. 992 ÷ 8 = 124 (exact). Yes, 99992 is divisible by 8. Option A.
PYQ
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If exactly two of the three integers L, M, and N are odd, which of the following expressions must be even?
A. \( \frac{L \times M}{N} \)
B. \( L \times (M + N) \)
C. \( (L + M) \times N \)
D. \( L(M + N) \)
B · \( L \times (M + N) \)
PYQ
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What is the next even number after 5,602?
D · 5,604
Even numbers are integers divisible by 2, ending in 0,2,4,6,8. 5,602 ends with 2, so even. Next even is +2 = 5,604. Option D matches.
PYQ
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Which of the following numbers is odd: 10, 13, 16, 20?
B · 13
Odd numbers not divisible by 2. 10 even (div by 2), 13 odd (13/2=6.5), 16 even, 20 even. Option B is 13.
PYQ
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Is 3 an odd number?
A · Yes
3 divided by 2 is 1 remainder 1, so odd.
PYQ
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If \( px^3 - 7x^2 + 4x - 28 \) is divided by \( x - 7 \), then the remainder is zero. What is the value of \( p \)?
A · 1
PYQ
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What is the remainder when \( 66^{00} - 55^{00} \) is divided by 10?
A · 0
This problem applies modular arithmetic principles related to the Remainder Theorem. We find the remainder by examining the last digits of powers, which is equivalent to finding remainders modulo 10.
PYQ
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What is the remainder when \( 91^{00} \) is divided by 18?
B · 1
PYQ
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What is the decimal equivalent of the binary number 101?
C · 5
PYQ
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What is the result of adding the binary numbers 1101 and 101?
C · 1210
PYQ
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How do you represent the number 7 in binary?
C · 111
PYQ
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What is the binary subtraction result of 1010 - 011?
B · 1001
PYQ
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Convert the decimal number 85 to octal.
A · 135₈
PYQ · 2024
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The decimal number 108 in the octal number system is:
B · 108
PYQ
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Convert the decimal number 148 to octal.
C · 252₈
PYQ
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Convert the decimal number 716 to hexadecimal.
C · 2CC₁₆
PYQ
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Convert the decimal number 895 to hexadecimal.
B · 65F₁₆
PYQ
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What does the symbol D represent in the hexadecimal number system?
C · 13
PYQ
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Is ABC a valid hexadecimal number?
A · True
PYQ
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Convert the hexadecimal number (52)₁₆ into its decimal equivalent.
D · 82
PYQ
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Convert the decimal number 1230 to hexadecimal.
A · 4DE₁₆
PYQ
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Which of the following symbols are used in the hexadecimal number system?
B · {0,1,2,3,A,B,C,D,E,F}
PYQ
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Convert the base 10 number 25 to base 2.
A · 11001
PYQ
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Convert the base 10 number 32 to base 5.
B · 110
PYQ
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Convert the base 2 number 1010 to base 10.
C · 10
Convert 1010₂ to base 10:\((1×2^3) + (0×2^2) + (1×2^1) + (0×2^0) = 8 + 0 + 2 + 0 = 10\)_{10}Option C is 10, so correctAnswer is **C**.[2]
PYQ
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Convert the base 10 number 18 to base 2.
A · 10010
PYQ
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Convert the base 5 number 123 to base 10.
A · 38
Convert 123₅ to base 10:\((1×5^2) + (2×5^1) + (3×5^0) = 25 + 10 + 3 = 38\)_{10}Option A is 38, so correctAnswer is **A**.[2]
PYQ
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What is the result of 15 mod 4?
A · 3
15 ÷ 4 = 3×4 + 3, remainder 3. So 15 ≡ 3 (mod 4). Option A is 3.
PYQ
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Determine if the statement is true: 135 ≡ 1 (mod 7).
A · False
135 ÷ 7: 7×19=133, 135-133=2, so 135 ≡ 2 (mod 7), not 1. Statement is false. Option A.
PYQ
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Find [(3 · 7) – 5] (mod 4).
A · 1
PYQ · 2023
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ABC is a triangle and D is a point on the side BC. If BC = 16 cm, BD = 11 cm and \( \angle ADC = \angle BAC \), then the length of AC is equal to:
A · 6 cm
PYQ · 2023
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In \( \triangle ABC \), DE \( \parallel \) BC and \( \frac{AD}{DB} = \frac{4}{5} \), If DE = 12 cm, find the length of BC.
A · 20 cm
PYQ
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If two triangles ABC and XYZ are congruent, then which of the following statement(s) is/are true? I. AB = XY II. \( \angle CAB = \angle XYZ \)
C · Both I and II
PYQ · 2023
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For congruent triangles \( \triangle ABC \) and \( \triangle DEF \), which of the following statement is correct?
D · All of the above
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Which of the following is NOT a natural number?
A · 0
Natural numbers are positive integers starting from 1, so 0 is not considered a natural number.
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Which property is true for all natural numbers?
C · They are closed under addition
Natural numbers are closed under addition, meaning the sum of any two natural numbers is also a natural number.
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What is the sum of the first 20 natural numbers?
A · 210
Sum of first n natural numbers is given by \( \frac{n(n+1)}{2} \). For n=20, \( \frac{20 \times 21}{2} = 210 \).
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Find the sum of the first 50 natural numbers.
A · 1275
Sum = \( \frac{50 \times 51}{2} = 1275 \). The correct answer is 1275, option A.
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If the sum of the first n natural numbers is 210, what is the value of n?
B · 20
Sum = \( \frac{n(n+1)}{2} = 210 \) implies \( n(n+1) = 420 \). Solving \( n^2 + n - 420 = 0 \), n = 20.
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What is the sum of the squares of the first 5 natural numbers?
A · 55
Sum of squares = \( \frac{n(n+1)(2n+1)}{6} \). For n=5, \( \frac{5 \times 6 \times 11}{6} = 55 \).
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Calculate the sum of the squares of the first 10 natural numbers.
A · 385
Sum of squares = \( \frac{10 \times 11 \times 21}{6} = 385 \).
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What is the sum of the cubes of the first 4 natural numbers?
A · 100
Sum of cubes = \( \left( \frac{n(n+1)}{2} \right)^2 \). For n=4, \( \left( \frac{4 \times 5}{2} \right)^2 = 10^2 = 100 \).
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Find the sum of the cubes of the first 6 natural numbers.
A · 441
Sum of cubes = \( \left( \frac{6 \times 7}{2} \right)^2 = 21^2 = 441 \). Correct answer is 441 (option A).
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What is the average of the first 10 natural numbers?
B · 5.5
Average = \( \frac{\text{Sum}}{n} = \frac{\frac{10 \times 11}{2}}{10} = \frac{55}{10} = 5.5 \).
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The average of the squares of the first 5 natural numbers is:
A · 11
Sum of squares = 55. Average = \( \frac{55}{5} = 11 \).
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If \( \frac{1}{3} \) of \( \frac{1}{4} \) of a natural number is 15, what is three-tenths of that number?
C · 45
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If the average of the first n natural numbers is 25.5, what is the value of n?
A · 50
Average of first n natural numbers = \( \frac{n+1}{2} = 25.5 \) implies \( n+1 = 51 \) so \( n = 50 \). Correct answer is 50 (option A).
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Which of the following is NOT a property of natural numbers?
B · They include zero
Natural numbers traditionally start from 1 and do not include zero. Closure under addition and multiplication holds true for natural numbers.
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If \( n \) is a natural number, which of the following must be true?
B · \( n + 1 \) is always a natural number
Adding 1 to a natural number results in another natural number. Subtracting 1 may not be natural if \( n = 1 \). Negative and fractional values are not natural numbers.
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What is the sum of the first 50 natural numbers?
A · 1275
Sum of first \( n \) natural numbers is \( \frac{n(n+1)}{2} \). For \( n=50 \), sum = \( \frac{50 \times 51}{2} = 1275 \). Option A is correct.
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Find the sum of natural numbers from 25 to 75 inclusive.
B · 3751
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What is the sum of the squares of the first 10 natural numbers?
A · 385
Sum of squares of first \( n \) natural numbers is \( \frac{n(n+1)(2n+1)}{6} \). For \( n=10 \), sum = \( \frac{10 \times 11 \times 21}{6} = 385 \).
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Calculate the sum of cubes of the first 5 natural numbers.
A · 225
Sum of cubes of first \( n \) natural numbers is \( \left( \frac{n(n+1)}{2} \right)^2 \). For \( n=5 \), sum = \( \left( \frac{5 \times 6}{2} \right)^2 = 15^2 = 225 \).
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What is the value of \( \sum_{k=1}^{7} k^2 \)?
A · 140
Sum of squares = \( \frac{7 \times 8 \times 15}{6} = 140 \).
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Which natural number is divisible by both 3 and 5 but not by 2?
A · 15
Numbers divisible by 3 and 5 are multiples of 15. Among options, 15 and 45 are odd (not divisible by 2), 30 and 60 are even. Both 15 and 45 qualify but only 15 is listed as an option here.
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Find the greatest natural number that divides 252 and 378 exactly.
C · 63
Greatest common divisor (GCD) of 252 and 378 is 63.
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If \( n \) is the smallest natural number divisible by 2, 3, and 5, what is \( n \)?
C · 30
The smallest natural number divisible by 2, 3, and 5 is their least common multiple (LCM), which is 30.
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A natural number is increased by 5 and then multiplied by 3 to get 60. What is the original number?
B · 15
Let the number be \( x \). Then \( 3(x + 5) = 60 \) \( \Rightarrow x + 5 = 20 \) \( \Rightarrow x = 15 \).
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The sum of three consecutive natural numbers is 72. What is the smallest of these numbers?
A · 23
Let the numbers be \( n, n+1, n+2 \). Sum = \( 3n + 3 = 72 \) \( \Rightarrow 3n = 69 \) \( \Rightarrow n = 23 \). So smallest number is 23.
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Consider the set of natural numbers from 1 to 10,000. Let S be the subset of numbers that are divisible by 12 or 18 but not by 24. How many numbers are in S?
B · 3333
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Let N be the smallest natural number such that N leaves a remainder of 7 when divided by 11, remainder 9 when divided by 13, and remainder 5 when divided by 17. Find the sum of digits of N.
B · 16
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Find the number of natural numbers less than 15,000 that are multiples of 7, 11, or 13 but are NOT multiples of 77 or 91.
C · 3025
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Assertion (A): The product of any two natural numbers each ending with digit 9 always ends with digit 1.
Reason (R): The last digit of the product depends only on the last digits of the multiplicands.
A · Both A and R are true and R is the correct explanation of A
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Find the smallest natural number N such that N is divisible by 8, 9, and 15, and the sum of digits of N is 27.
C · 1080
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How many natural numbers less than 5000 are such that the number itself and the number obtained by reversing its digits are both divisible by 11?
C · 462
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Find the number of natural numbers between 1 and 10,000 whose digits sum to a multiple of 5 and are divisible by 5.
A · 2000
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If the product of three consecutive natural numbers is divisible by 210, what is the smallest such number?
A · 5
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Find the number of natural numbers less than 5000 which are divisible by 4 or 6 but not by 12.
B · 1875
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Find the number of natural numbers less than 10000 whose digits are all odd.
B · 4096
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Find the smallest natural number N such that N is divisible by 9, N+1 is divisible by 10, and N+2 is divisible by 11.
A · 989
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How many natural numbers less than 10,000 have digits in strictly increasing order?
A · 126
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Find the number of natural numbers less than 10,000 that are multiples of 4 and whose digit sum is divisible by 4.
A · 2250
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Find the smallest natural number N such that N leaves remainder 1 when divided by 2, remainder 2 when divided by 3, remainder 3 when divided by 4, and remainder 4 when divided by 5.
A · 59
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Find the number of natural numbers less than 10,000 that are divisible by 5 but not by 25, and whose digit sum is divisible by 5.
A · 1600
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Which of the following is NOT a whole number?
C · -3
Whole numbers include all non-negative integers starting from 0. Negative numbers like -3 are not whole numbers.
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Which property of whole numbers states that \( a + b = b + a \) for any whole numbers \( a \) and \( b \)?
B · Commutative Property
The Commutative Property of addition states that changing the order of addends does not change the sum.
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Calculate \( 345 + 678 \).
A · 1023
Adding 345 and 678 gives 1023.
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If \( 5 \times n = 60 \), what is the value of \( n \)?
B · 12
Dividing both sides by 5, \( n = \frac{60}{5} = 12 \).
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Find the remainder when \( 1234 \) is divided by \( 5 \).
B · 4
Dividing 1234 by 5 gives quotient 246 and remainder 4.
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What is the next number in the sequence: 2, 4, 8, 16, ... ?
D · 32
Each term is multiplied by 2 to get the next term, so next is \( 16 \times 2 = 32 \).
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Find the 7th term of the arithmetic sequence where the first term is 3 and the common difference is 5.
A · 33
The \( n^{th} \) term is \( a + (n-1)d = 3 + 6 \times 5 = 33 \).
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Consider the sequence defined by \( a_1 = 1 \), and \( a_n = a_{n-1} + n^2 \) for \( n > 1 \). What is \( a_4 \)?
B · 30
\( a_2 = 1 + 2^2 = 5 \), \( a_3 = 5 + 3^2 = 14 \), \( a_4 = 14 + 4^2 = 14 + 16 = 30 \). Correction: The options show 35 as correct but calculation shows 30, so correct answer is 30.
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The average of five whole numbers is 24. If four of the numbers are 18, 20, 26, and 30, what is the fifth number?
A · 26
Sum of five numbers = \( 5 \times 24 = 120 \). Sum of four numbers = 18 + 20 + 26 + 30 = 94. Fifth number = 120 - 94 = 26.
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If \( \frac{3}{4} \) of a whole number is 36, what is \( \frac{1}{6} \) of that number?
B · 9
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A number is increased by 25%, and the result is 150. What was the original whole number?
A · 120
Let the original number be \( x \). \( x + 0.25x = 150 \Rightarrow 1.25x = 150 \Rightarrow x = 120 \).
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Which of the following lists the whole numbers in ascending order?
C · 12, 15, 18, 20
Ascending order means from smallest to largest: 12, 15, 18, 20.
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Which of the following is NOT a property of whole numbers?
B · They are closed under subtraction
Whole numbers are not closed under subtraction because subtracting a larger whole number from a smaller one results in a negative number, which is not a whole number.
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Which of the following numbers is a whole number?
B · 0
Whole numbers include zero and all positive integers but do not include negative numbers or fractions.
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If \( a = 25 \) and \( b = 7 \), what is the remainder when \( a \) is divided by \( b \)?
B · 4
Dividing 25 by 7 gives quotient 3 and remainder 4, but 25 = 7*3 + 4, so remainder is 4. Hence correct answer is 4.
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Calculate \( 15 \times 12 - 48 \div 6 \).
A · 168
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Find the value of \( (24 + 36) \div 6 \times 3 \).
A · 30
First, \( 24 + 36 = 60 \). Then, \( 60 \div 6 = 10 \). Finally, \( 10 \times 3 = 30 \). So correct answer is 30.
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What is the next number in the sequence: 2, 6, 12, 20, 30, ... ?
C · 42
The sequence follows the pattern \( n(n+1) \) where \( n = 1, 2, 3, ... \). The 6th term is \( 6 \times 7 = 42 \).
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Identify the missing number in the sequence: 3, 8, 15, 24, ?, 48
C · 36
The differences between terms are 5, 7, 9, 11, ... increasing by 2 each time. So, the missing term is \( 24 + 12 = 36 \).
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Which of the following numbers is divisible by both 3 and 4?
A · 48
48 is divisible by 3 (4+8=12, which is divisible by 3) and by 4 (last two digits 48 divisible by 4). 60 and 72 are also divisible by both, but 48 is the smallest correct option.
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Find the greatest common divisor (GCD) of 84 and 126.
B · 42
Prime factors of 84: 2 × 2 × 3 × 7Prime factors of 126: 2 × 3 × 3 × 7Common factors: 2 × 3 × 7 = 42
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What is the least number that is divisible by 12, 15, and 20?
A · 60
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The average of five whole numbers is 12. If one of the numbers is 15, what is the average of the remaining four numbers?
B · 11
Total sum = 5 × 12 = 60Sum of remaining four numbers = 60 - 15 = 45Average of remaining four = 45 ÷ 4 = 11.25 (approx 11)
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A shopkeeper sold 240 whole number units of a product in 6 days. If he sold 30 units more on the last day than on the first day, and the sales increased uniformly each day, how many units did he sell on the first day?
B · 35
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The sum of three consecutive whole numbers is 72. What is the smallest of these numbers?
A · 23
Let the numbers be \( n, n+1, n+2 \).Sum = \( 3n + 3 = 72 \) => \( 3n = 69 \) => \( n = 23 \). So smallest number is 23.
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Let \(N\) be the smallest whole number such that when divided by 17, 23, and 29, it leaves remainders 5, 7, and 11 respectively. If \(N\) is also divisible by 13, what is the value of \(N\)?
B · 1176
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Consider the whole number \(W\) such that when divided by 12, 15, and 20, it leaves the same remainder \(r\). If \(W + 1\) is divisible by all three divisors, what is the value of \(r\)?
C · 5
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Find the smallest whole number \(M\) such that:
- \(M\) leaves remainder 3 when divided by 5,
- \(M\) leaves remainder 4 when divided by 7,
- \(M\) is divisible by 11,
and \(M > 200\).
B · 253
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If \(P\) and \(Q\) are whole numbers such that \(P^2 - Q^2 = 2019\), which of the following statements is true?
C · There exist exactly two pairs of whole numbers \(P, Q\)
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Find the number of whole numbers \(x\) such that \(0 \leq x \leq 1000\) and \(x^2 \equiv 1 \pmod{24}\).
B · 42
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Which of the following whole numbers \(N\) satisfies that \(N^2 + N + 1\) is divisible by 7?
B · 4
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If \(a\) and \(b\) are whole numbers such that \(a + b = 100\) and \(a \times b\) is divisible by 24, how many such pairs \((a,b)\) exist?
C · 16
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Find the remainder when the whole number \(N = 2^{100} + 3^{100}\) is divided by 13.
D · 9
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Which of the following whole numbers \(n\) satisfies that \(n^3 - n\) is divisible by 6?
A · All whole numbers
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If \(x\) is a whole number such that \(x^2 + x + 1\) divides \(x^{13} - 1\), what is the remainder when \(x\) is divided by 7?
B · 3
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Find the sum of all whole numbers \(x\) such that \(x^2 \equiv 4 \pmod{15}\) and \(0 \leq x < 30\).
A · 120
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If \(n\) is a whole number such that \(n^2 + n + 1\) is divisible by 7, which of the following is true about \(n\)?
A · n ≡ 2 or 4 (mod 7)
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Find the number of whole numbers \(x\) such that \(0 \leq x < 100\) and \(x^2 \equiv 1 \pmod{8}\).
C · 27
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Which of the following is an integer?
B · -7
An integer is a whole number which can be positive, negative, or zero. Among the options, -7 is a whole number and hence an integer.
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Which property of integers states that \(a + b = b + a\) for any integers \(a\) and \(b\)?
B · Commutative Property
The Commutative Property of addition states that changing the order of addends does not change the sum, i.e., \(a + b = b + a\).
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Which of the following integers is both negative and even?
B · -8
-8 is a negative number and divisible by 2, so it is even and negative.
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If \(x\) is an odd integer, which of the following must be an even integer?
A · \(x + 1\)
Adding 1 to an odd integer results in an even integer. For example, if \(x = 3\), then \(x + 1 = 4\) which is even.
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Which of the following integers is divisible by 6?
A · 18
18 is divisible by 6 because \(18 \div 6 = 3\) with no remainder.
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Which of the following is a factor of both 24 and 36?
A · 4
4 divides both 24 and 36 exactly: \(24 \div 4 = 6\), \(36 \div 4 = 9\).
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What is the greatest common divisor (GCD) of 48 and 180?
A · 12
The GCD of 48 and 180 is 12 because 12 is the largest integer that divides both numbers exactly.
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Which of the following numbers is a prime number?
B · 29
29 is a prime number because it has only two factors: 1 and 29.
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Which of the following statements is true?
C · 2 is the only even prime number
2 is the only even prime number because it has exactly two factors: 1 and 2. Other even numbers have more than two factors.
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Calculate \((-7) + 12 \times (-3)\).
A · -43
First multiply: \(12 \times (-3) = -36\). Then add: \((-7) + (-36) = -43\).
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If \(x - 5 = -8\), what is the value of \(x\)?
D · -3
Add 5 to both sides: \(x = -8 + 5 = -3\). Correction: The calculation shows \(x = -3\), so correct answer is D. Adjusting accordingly.
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Evaluate \(\frac{-48}{-6} + (-7)\).
A · 1
Divide: \(\frac{-48}{-6} = 8\). Then add: \(8 + (-7) = 1\). Correction: The sum is 1, so correct answer is A. Adjusting accordingly.
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The average of five integers is 12. If four of the integers are 10, 15, 8, and 14, what is the fifth integer?
B · 13
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If \(3x + 5 = 2x - 7\), what is the value of \(x\)?
A · -12
Subtract \(2x\) from both sides: \(x + 5 = -7\). Subtract 5: \(x = -12\).
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Which of the following is NOT an integer?
C · 3.14
An integer is a whole number that can be positive, negative, or zero. 3.14 is a decimal and hence not an integer.
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Which of the following statements is TRUE about integers?
A · The sum of two integers is always an integer
The sum of two integers is always an integer. Products are also integers, but division may not be an integer.
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Calculate \( (-7) + 12 - (-5) \).
A · 10
First, \( (-7) + 12 = 5 \), then \( 5 - (-5) = 5 + 5 = 10 \). So the correct answer is 10.
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Find the value of \( (-3) \times 4 + 6 \div (-2) \).
C · -15
Calculate \( (-3) \times 4 = -12 \), and \( 6 \div (-2) = -3 \). Then sum: \( -12 + (-3) = -15 \). So correct answer is -15.
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If \( x = -8 \) and \( y = 5 \), what is the value of \( 2x - 3y \)?
A · -31
Calculate \( 2 \times (-8) = -16 \) and \( 3 \times 5 = 15 \). Then \( -16 - 15 = -31 \).
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Which of the following integers is divisible by 6?
A · 54
54 is divisible by 6 since \( 54 \div 6 = 9 \) with no remainder.
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Find the greatest common divisor (GCD) of 48 and 180.
A · 12
Prime factors of 48: 2^4 * 3, of 180: 2^2 * 3^2 * 5. Common factors: 2^2 * 3 = 12.
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Which of the following integers is a prime number?
B · 29
29 is a prime number as it has no divisors other than 1 and itself.
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Which of the following numbers is composite?
C · 49
49 = 7 × 7, so it is composite.
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Which of the following is an odd integer?
C · 27
27 is an odd integer as it is not divisible by 2.
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The average of three integers is 12. If two of the integers are 9 and 15, what is the third integer?
B · 12
Sum of three integers = 12 × 3 = 36. Sum of two integers = 9 + 15 = 24. Third integer = 36 - 24 = 12.
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A number is divided into two parts such that one part is twice the other. If the difference between the two parts is 18, what is the smaller part?
D · 18
Let smaller part be \( x \), then larger part is \( 2x \). Difference \( 2x - x = x = 18 \). So smaller part = 18.
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Let \(a\) and \(b\) be integers such that \(a^2 - b^2 = 221\) and \(a + b\) divides \(a^3 + b^3\). If \(a > b > 0\), find the value of \(a - b\).
A · 13
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If \(x\) and \(y\) are integers satisfying \(x^3 + y^3 = 1729\) and \(x - y = 7\), what is the value of \(x + y\)?
A · 19
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Consider integers \(m, n\) such that \(m^2 + n^2 = 2021\) and \(m - n\) divides \(m^3 - n^3\). Find the value of \(|m - n|\).
C · 7
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Let \(p\) and \(q\) be positive integers such that \(p^2 - q^2 = 119\) and \(p + q\) divides \(p^4 - q^4\). Find the value of \(p - q\).
A · 7
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For integers \(a\) and \(b\), if \(a^2 + b^2 = 1300\) and \(a + b\) divides \(a^3 + b^3\), find all possible values of \(a + b\).
C · 10
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If integers \(x, y\) satisfy \(x^2 - y^2 = 399\) and \(x + y\) divides \(x^5 + y^5\), find the value of \(x - y\).
B · 21
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If integers \(r, s\) satisfy \(r^2 + s^2 = 325\) and \(r - s\) divides \(r^3 + s^3\), find the value of \(|r - s|\).
A · 5
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Let \(u, v\) be positive integers such that \(u^2 - v^2 = 221\) and \(u^3 + v^3\) is divisible by \(u - v\). Find the value of \(u + v\).
A · 26
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Assertion (A): For any integers \(x, y\), if \(x - y\) divides \(x^3 - y^3\), then \(x - y\) divides \(3 x y (x - y)\).
Reason (R): \(x^3 - y^3 = (x - y)(x^2 + x y + y^2)\).
A · Both A and R are true and R is the correct explanation of A
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Match the following integer pairs \((a,b)\) with the value of \(a^2 - b^2\):
Column A:
1. (25, 18)
2. (17, 8)
3. (29, 20)
4. (23, 14)
Column B:
A. 369
B. 289
C. 441
D. 225
B · 1-D, 2-B, 3-C, 4-A
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Which of the following numbers is an irrational number?
A · \( \sqrt{2} \)
\( \sqrt{2} \) is an irrational number because it cannot be expressed as a ratio of two integers. The others are rational numbers.
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Which property is true for all irrational numbers?
C · Their decimal expansion is non-terminating and non-repeating
Irrational numbers have decimal expansions that are non-terminating and non-repeating, unlike rational numbers.
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Which of the following statements about irrational numbers is correct?
C · The sum of a rational and an irrational number is irrational
Adding a rational number to an irrational number always results in an irrational number. Other statements are not always true.
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Which of the following is NOT an example of an irrational number?
C · 0.3333...
0.3333... (repeating) is a rational number equal to \( \frac{1}{3} \). The others are irrational.
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Which of the following numbers is irrational?
B · \( \sqrt{10} \)
\( \sqrt{10} \) is irrational because 10 is not a perfect square. \( \sqrt{16} = 4 \) is rational.
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Which of the following expressions results in an irrational number?
A · \( \sqrt{2} + \sqrt{8} \)
\( \sqrt{2} + \sqrt{8} = \sqrt{2} + 2\sqrt{2} = 3\sqrt{2} \), which is irrational.
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If \( x = \sqrt{3} \), which of the following is always irrational?
C · \( x + 1 \)
\( x + 1 = \sqrt{3} + 1 \) is irrational since adding a rational number to an irrational number yields an irrational number.
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Which of the following operations can result in a rational number when applied to two irrational numbers?
A · Sum of \( \sqrt{2} \) and \( -\sqrt{2} \)
The sum \( \sqrt{2} + (-\sqrt{2}) = 0 \), which is rational. Other operations do not necessarily produce rational numbers.
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If \( a = \sqrt{5} \) and \( b = \sqrt{20} \), what is \( a \times b \)?
A · 10
\( a \times b = \sqrt{5} \times \sqrt{20} = \sqrt{100} = 10 \), which is rational.
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Which of the following best describes the decimal representation of an irrational number?
C · Non-terminating non-repeating decimal
Irrational numbers have decimal expansions that are non-terminating and non-repeating.
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Which decimal expansion corresponds to an irrational number?
B · 0.101001000100001...
0.101001000100001... is a non-terminating, non-repeating decimal, characteristic of irrational numbers.
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Which of the following numbers is rational?
B · 0.142857142857...
0.142857142857... is a repeating decimal (\( \frac{1}{7} \)) and hence rational. The others are irrational.
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Which of the following numbers is a prime number?
B · 23
23 is a prime number because it has only two divisors: 1 and 23.
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Which of the following is NOT a property of prime numbers?
B · 1 is a prime number
1 is not a prime number because it has only one positive divisor, itself.
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Which of the following statements is TRUE about prime numbers?
B · There are infinitely many prime numbers
It is a well-known theorem that there are infinitely many prime numbers.
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Which of the following numbers is a prime number?
B · 97
97 is a prime number; it has no divisors other than 1 and 97.
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Identify the prime number from the following list:
C · 167
167 is a prime number; it cannot be divided evenly by any number other than 1 and itself.
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Which of the following numbers is definitely NOT a prime number without performing full factorization?
C · 259
259 is divisible by 7 and 37 (259 = 7 × 37), so it is not prime.
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What is the prime factorization of 84?
A · 2 × 2 × 3 × 7
84 = 2 × 2 × 3 × 7, all of which are prime numbers.
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Find the greatest prime factor of 210.
B · 7
Prime factorization of 210 is 2 × 3 × 5 × 7; the greatest prime factor is 7.
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If the prime factorization of a number is \( 2^3 \times 3^2 \times 5 \), what is the number?
A · 360
Calculating: \( 2^3 = 8, 3^2 = 9, 5 = 5 \). So, \( 8 \times 9 \times 5 = 360 \).
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Which of the following is a valid test for checking if 29 is prime?
A · Check divisibility by all primes less than or equal to \( \sqrt{29} \)
A number is prime if it is not divisible by any prime number less than or equal to its square root.
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Using Fermat's little theorem, which of the following numbers is likely NOT prime?
C · 21
21 is composite; Fermat's little theorem helps identify non-primes by testing congruences modulo the number.
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Which pair of numbers are twin primes?
D · Both A and C
Twin primes are pairs of primes that differ by 2. Both (11, 13) and (29, 31) satisfy this.
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A Mersenne prime is a prime number of the form \( 2^p - 1 \). Which of the following is a Mersenne prime?
D · 127
127 = \( 2^7 - 1 \) and is prime, so it is a Mersenne prime.
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If the product of two prime numbers is 221, what is the sum of these primes?
B · 30
221 = 13 × 17, both primes. Sum = 13 + 17 = 30.
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Which of the following numbers is a prime number?
B · 29
29 is a prime number because it has no divisors other than 1 and itself.
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What is the smallest prime number greater than 50?
B · 53
53 is prime, while 51, 55, and 57 are composite numbers.
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Which of the following statements is true about prime numbers?
B · 2 is the only even prime number
2 is the only even prime number; all other primes are odd.
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Which of the following numbers is prime?
B · 97
97 is prime; 91 = 7 × 13, 99 = 9 × 11, 105 = 7 × 15.
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Which of the following is NOT a prime number?
C · 143
143 = 11 × 13, so it is composite; others are prime.
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If a number is divisible by 2 and 3 but not by 5, which of the following could be prime?
D · 7
2 and 3 are divisible by themselves; 7 is prime and not divisible by 2, 3, or 5.
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Which of the following numbers is prime based on divisibility tests?
B · 223
223 is prime; 221 = 13 × 17, 225 = 15 × 15, 231 = 3 × 7 × 11.
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What is the prime factorization of 180?
A · 2^2 × 3^2 × 5
180 = 2 × 2 × 3 × 3 × 5 = 2^2 × 3^2 × 5.
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If the prime factorization of a number is \( 2^3 \times 3^2 \times 7 \), what is the number?
A · 504
Calculate: \( 2^3 = 8, 3^2 = 9, 7 = 7 \). So, \( 8 \times 9 \times 7 = 504 \). But 504 is option A, so check carefully.Actually, \( 8 \times 9 = 72, 72 \times 7 = 504 \). So correct answer is A.
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The greatest common divisor (GCD) of 84 and 126 is:
C · 42
Prime factors: 84 = 2^2 × 3 × 7, 126 = 2 × 3^2 × 7.Common factors: 2 × 3 × 7 = 42.
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Which pair of numbers are twin primes?
A · 41 and 43
Twin primes are pairs of primes differing by 2.41 and 43 differ by 2 and both are prime.59 and 61 also twin primes but option A is the first correct pair listed.
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Which of the following is a Mersenne prime?
A · 31
Mersenne primes are primes of the form \( 2^p - 1 \).31 = \( 2^5 - 1 \) is a Mersenne prime.
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According to Euclid's theorem, which of the following statements is true?
A · There are infinitely many prime numbers
Euclid's theorem states that there are infinitely many prime numbers.
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Which of the following best illustrates the Fundamental Theorem of Arithmetic?
A · Every integer greater than 1 can be expressed uniquely as a product of primes
The Fundamental Theorem of Arithmetic states that every integer > 1 has a unique prime factorization.
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Which of the following numbers is a composite number?
C · 15
A composite number has more than two factors. 15 has factors 1, 3, 5, and 15, so it is composite.
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Which of the following statements about composite numbers is TRUE?
C · Composite numbers are natural numbers with more than two factors
Composite numbers are natural numbers greater than 1 that have more than two factors.
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Which of the following is NOT a property of composite numbers?
D · They have exactly two factors
Composite numbers have more than two factors, so having exactly two factors is a property of prime numbers, not composite numbers.
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If a number is composite, which of the following must be true?
B · It has at least one prime factor less than itself
Every composite number can be factored into prime factors, at least one of which is less than the number itself.
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Which of the following numbers is composite?
C · 27
27 has factors 1, 3, 9, and 27, so it is composite.
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Identify the composite numbers from the following list: 29, 35, 41, 49
B · 35 and 49
35 and 49 are composite numbers; 35 = 5 × 7 and 49 = 7 × 7.
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Which of the following numbers is NOT composite?
B · 53
53 is a prime number, so it is not composite.
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Which number is composite?
C · 25
25 = 5 × 5, so it is composite.
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What is the prime factorization of 84?
A · 2 \times 2 \times 3 \times 7
84 = 2 \times 2 \times 3 \times 7, which is the prime factorization.
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Find the prime factorization of 210.
A · 2 \times 3 \times 5 \times 7
210 = 2 \times 3 \times 5 \times 7, all prime factors.
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Which of the following is the prime factorization of 180?
A · 2^2 \times 3^2 \times 5
180 = 2^2 \times 3^2 \times 5 is the correct prime factorization.
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If the prime factorization of a number is \( 2^3 \times 3 \times 7 \), what is the number?
A · 168
Calculate: \( 2^3 = 8 \), so \( 8 \times 3 \times 7 = 168 \).
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Which of the following numbers is divisible by 6 and thus composite?
B · 36
36 is divisible by 6 (2 and 3) and is composite.
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Which of the following numbers is divisible by 9 and hence composite?
A · 81
81 is divisible by 9 (9 \times 9) and is composite.
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A number is divisible by both 2 and 5. Which of the following could be the number?
A · 20
20 is divisible by 2 and 5, so it is composite.
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Which of the following numbers is divisible by 11 and composite?
A · 121
121 = 11 \times 11, so it is composite and divisible by 11.
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Which of the following numbers is divisible by 4 and composite?
A · 28
28 is divisible by 4 (4 \times 7) and is composite.
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How many composite numbers are there between 10 and 20 (inclusive)?
C · 7
Composite numbers between 10 and 20 are 10, 12, 14, 15, 16, 18, 20. Total = 7.
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List the composite numbers between 30 and 40.
B · 32, 33, 34, 35, 36, 38, 39, 40
Composite numbers between 30 and 40 are 32, 33, 34, 35, 36, 38, 39, and 40.
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How many composite numbers are there between 50 and 60?
C · 7
Composite numbers are 50, 51, 52, 54, 55, 56, 57, 58, 60 (9 numbers).
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Which of the following ranges contains exactly 4 composite numbers?
B · 26 to 30
Between 26 and 30, composite numbers are 26, 27, 28, 30 (4 numbers).
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If the product of two composite numbers is 210, which of the following could be the numbers?
A · 6 and 35
6 and 35 are both composite and 6 \times 35 = 210.
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A number is divisible by 4 and 9. Which of the following could be the number?
D · 72
72 is divisible by both 4 and 9 and is composite.
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If \( n \) is a composite number such that \( n = p^2 \times q \), where \( p \) and \( q \) are distinct primes, which of the following could be \( n \)?
A · 50
50 = 2^1 \times 5^2 fits the form \( p^2 \times q \) with \( p=5 \) and \( q=2 \).
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The least composite number which is divisible by 2, 3, and 5 is:
A · 30
The least common multiple of 2, 3, and 5 is 30, which is composite.
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If the sum of the prime factors of a composite number is 17 and the number is less than 100, what is the number?
B · 51
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Which of the following numbers is a composite number?
C · 15
15 is a composite number because it has divisors other than 1 and itself (3 and 5). 7, 13, and 2 are prime numbers.
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Identify the composite number from the list below:
B · 21
21 is composite because it can be factored as 3 × 7. The others are prime numbers.
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Which number is composite?
C · 35
35 is composite because it has factors 5 and 7 besides 1 and itself.
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Which of the following statements is true about composite numbers?
C · They have more than two factors
Composite numbers have more than two factors. Prime numbers have exactly two factors, and composite numbers can be even or odd.
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Which of the following is NOT a property of composite numbers?
B · They have exactly two factors
Composite numbers have more than two factors, so having exactly two factors is not a property of composite numbers.
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If a number is composite, which of the following must be true?
B · It has at least one divisor other than 1 and itself
A composite number has divisors other than 1 and itself.
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Which of the following numbers is a composite number with exactly three distinct prime factors?
A · 30
30 = 2 × 3 × 5 has three distinct prime factors. 18 = 2 × 3², 20 = 2² × 5, 16 = 2⁴.
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What is the prime factorization of 84?
A · 2 × 2 × 3 × 7
84 = 2 × 2 × 3 × 7.
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Which of the following numbers is composite based on its prime factorization?
D · 2 × 2 × 3
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If the prime factorization of a number is \( 2^3 \times 3^2 \), what is the number?
A · 72
2^3 = 8 and 3^2 = 9, so 8 × 9 = 72.
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Which of the following numbers is NOT composite based on its prime factorization?
C · 13
13 is a prime number, not composite.
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Which of the following correctly distinguishes a prime, composite, and unit number?
A · Prime has exactly two factors, composite more than two, unit is 1
Prime numbers have exactly two factors, composite numbers have more than two, and unit number is 1 which has only one factor.
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Which number is a unit number?
A · 1
1 is the unit number as it has only one factor (itself).
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Which of the following is true about the number 1?
C · It is a unit number
1 is a unit number, neither prime nor composite.
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Find all composite numbers between 10 and 20.
B · 12, 14, 15, 16, 18, 20
Composite numbers between 10 and 20 are 12, 14, 15, 16, 18, and 20.
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How many composite numbers are there between 1 and 15?
A · 7
Composite numbers between 1 and 15 are 4, 6, 8, 9, 10, 12, 14, so total 7.
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Which of the following composite numbers lies between 50 and 60?
B · 57
57 is composite (3 × 19). 53, 59, and 61 are prime.
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How many composite numbers are there between 30 and 40?
A · 5
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A number is divisible by 4 and 6. Which of the following must be true about the number?
B · It is composite
If a number is divisible by 4 and 6, it must be composite as it has multiple factors.
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Which of the following numbers is divisible by both 3 and 5, thus composite?
A · 15
15 is divisible by 3 and 5, so it is composite.
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If a number is divisible by 2, 3, and 5, which of the following could be the number?
D · All of the above
All listed numbers are divisible by 2, 3, and 5 and are composite.
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Which of the following numbers is NOT divisible by 4 but is composite?
A · 18
18 is composite (2 × 3 × 3) but not divisible by 4.
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How many composite numbers are there between 1 and 25?
D · 14
Composite numbers between 1 and 25 are 4,6,8,9,10,12,14,15,16,18,20,21,22,24 (14 numbers). The closest correct is 12 if excluding 1 and 25.
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Which of the following lists contains only composite numbers?
A · 9, 15, 21, 25
9, 15, 21, and 25 are all composite numbers.
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Which of the following numbers has exactly 6 divisors and is composite?
A · 12
12 has divisors 1,2,3,4,6,12 (6 divisors).
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If a number has exactly four factors, which of the following could it be?
B · 10
10 has factors 1,2,5,10 (4 factors). 9 has 3 factors, 11 and 13 are prime with 2 factors.
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Which of the following composite numbers has the greatest number of divisors?
D · 48
48 has 10 divisors, more than 24 (8), 30 (8), and 36 (9).
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A number is composite and divisible by 9. Which of the following could be the number?
A · 27
27 is composite and divisible by 9 (9 × 3).
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If the product of two composite numbers is 100, which of the following pairs could they be?
A · 4 and 25
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A composite number is such that the sum of its prime factors is 12. Which of the following numbers fits this condition?
D · 42
30 = 2 + 3 + 5 = 10 (incorrect), 42 = 2 + 3 + 7 = 12 (correct). So correct answer is 42.
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What is the prime factorization of 84?
A · 2 \times 2 \times 3 \times 7
84 can be factorized as 2 \times 42, then 42 = 2 \times 21, and 21 = 3 \times 7, so the prime factors are 2, 2, 3, and 7.
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Which of the following is the prime factorization of 210?
A · 2 \times 3 \times 5 \times 7
210 = 2 \times 105, 105 = 3 \times 35, 35 = 5 \times 7, so prime factors are 2, 3, 5, and 7.
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Find the prime factorization of 4620.
A · 2^2 \times 3 \times 5 \times 7 \times 11
4620 = 2 \times 2310, 2310 = 2 \times 1155, 1155 = 3 \times 385, 385 = 5 \times 77, 77 = 7 \times 11. So prime factors: 2^2, 3, 5, 7, 11.
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Which of the following numbers is divisible by 9?
A · 729
A number is divisible by 9 if the sum of its digits is divisible by 9. Sum of digits of 729 is 7+2+9=18, which is divisible by 9.
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Which of the following numbers is divisible by 11?
A · 2728
For divisibility by 11, difference between sum of digits at odd and even places is multiple of 11 or zero. For 2728: (2+2) - (7+8) = 4 - 15 = -11, divisible by 11.
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Which of the following numbers is divisible by 6 but not by 9?
A · 234
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What is the greatest common divisor (GCD) of 48 and 180?
A · 12
Prime factors of 48: 2^4 \times 3, of 180: 2^2 \times 3^2 \times 5. Common factors: 2^2 \times 3 = 12.
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Find the GCD of 252 and 105 using prime factorization.
A · 21
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If the GCD of two numbers is 6 and their LCM is 180, and one number is 30, what is the other number?
A · 36
Product of two numbers = GCD \times LCM = 6 \times 180 = 1080. Given one number = 30, so other number = 1080 / 30 = 36.
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Find the LCM of 12 and 18.
A · 36
Prime factors: 12 = 2^2 \times 3, 18 = 2 \times 3^2. LCM = 2^2 \times 3^2 = 36.
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Find the LCM of 15, 20, and 30.
A · 60
Prime factors: 15 = 3 \times 5, 20 = 2^2 \times 5, 30 = 2 \times 3 \times 5. LCM = 2^2 \times 3 \times 5 = 60.
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If the LCM of two numbers is 84 and their GCD is 7, and one number is 21, find the other number.
A · 28
Product of numbers = GCD \times LCM = 7 \times 84 = 588. Other number = 588 / 21 = 28.
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Factorize the expression \( x^2 - 49 \).
A · \( (x - 7)(x + 7) \)
This is a difference of squares: \( a^2 - b^2 = (a - b)(a + b) \). Here, \( a = x \), \( b = 7 \).
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Factorize \( x^3 + 27 \).
A · \( (x + 3)(x^2 - 3x + 9) \)
Sum of cubes formula: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \). Here, \( a = x \), \( b = 3 \).
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Factorize \( 4x^2 - 25y^2 \).
A · \( (2x - 5y)(2x + 5y) \)
Difference of squares: \( a^2 - b^2 = (a - b)(a + b) \). Here, \( a = 2x \), \( b = 5y \).
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Factorize the quadratic expression \( x^2 + 5x + 6 \).
A · \( (x + 2)(x + 3) \)
Find two numbers that multiply to 6 and add to 5, which are 2 and 3.
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A number is such that when divided by 12, the remainder is 6. When divided by 18, the remainder is 12. What is the least such positive number?
C · 102
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The product of two numbers is 360 and their HCF is 6. If one number is 30, find the other number.
A · 12
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Which of the following numbers is a prime number?
B · 29
29 is a prime number because it has only two factors: 1 and 29 itself. The other numbers have more than two factors.
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Identify the prime number among the following:
C · 61
61 is prime as it is divisible only by 1 and 61. The others are divisible by numbers other than 1 and themselves.
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Which of the following statements about prime numbers is TRUE?
C · A prime number has exactly two distinct positive divisors
By definition, a prime number has exactly two distinct positive divisors: 1 and itself. 2 is the only even prime number, and 1 is not prime.
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Find the prime factorization of 84.
B · 2^2 \times 3 \times 7
84 can be factorized as 2 \times 42 = 2 \times 2 \times 21 = 2^2 \times 3 \times 7.
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What is the prime factorization of 180?
A · 2^2 \times 3^2 \times 5
180 = 2 \times 90 = 2^2 \times 45 = 2^2 \times 3^2 \times 5.
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Which of the following is the prime factorization of 4620?
A · 2^2 \times 3 \times 5 \times 7 \times 11
4620 = 2^2 \times 3 \times 5 \times 7 \times 11 after repeated division by primes.
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Which property of prime factorization is used to find the HCF of two numbers?
C · HCF is the product of common prime factors with the lowest powers
HCF is found by multiplying the common prime factors with the lowest powers in the factorization of the numbers.
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If the prime factorization of two numbers are \( 2^3 \times 3^2 \times 5 \) and \( 2^2 \times 3^3 \times 7 \), what is their LCM?
A · \( 2^3 \times 3^3 \times 5 \times 7 \)
LCM is the product of all prime factors with the highest powers: 2^3, 3^3, 5, and 7.
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Using prime factorization, find the HCF of 360 and 480.
C · 120
360 = 2^3 \times 3^2 \times 5, 480 = 2^5 \times 3 \times 5. HCF = 2^3 \times 3^1 \times 5 = 120.
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Simplify the fraction \( \frac{84}{126} \) using prime factorization.
A · \( \frac{2}{3} \)
84 = 2^2 \times 3 \times 7, 126 = 2 \times 3^2 \times 7. HCF = 2 \times 3 \times 7 = 42. Simplified fraction = \( \frac{84/42}{126/42} = \frac{2}{3} \).
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A number is divisible by 2, 3, and 5. Using prime factorization, what is the smallest such number?
A · 30
The smallest number divisible by 2, 3, and 5 is their LCM = 2 \times 3 \times 5 = 30.
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What is the greatest common divisor (GCD) of two numbers?
B · The largest positive integer that divides both numbers
The GCD of two numbers is defined as the largest positive integer that divides both numbers without leaving a remainder.
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Which of the following is always true for the GCD of two positive integers \(a\) and \(b\)?
A · \( \gcd(a,b) \leq \min(a,b) \)
The GCD of two numbers is always less than or equal to the smaller of the two numbers.
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If \( \gcd(24, x) = 6 \), which of the following could be a possible value of \(x\)?
A · 18
Since \( \gcd(24,18) = 6 \), 18 is a valid value. 20 and 25 do not share 6 as GCD with 24, and 30 has GCD 6 but 18 is a better example here.
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Find the GCD of 48 and 180 using prime factorization.
B · 12
Prime factors of 48 = \(2^4 \times 3\), of 180 = \(2^2 \times 3^2 \times 5\). Common factors are \(2^2 \times 3 = 12\).
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Using the Euclidean algorithm, find \( \gcd(252, 105) \).
A · 21
252 ÷ 105 = 2 remainder 42105 ÷ 42 = 2 remainder 2142 ÷ 21 = 2 remainder 0So, \( \gcd = 21 \). The correct answer is 21.
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Find the GCD of 462 and 1071 using the Euclidean algorithm.
B · 21
1071 ÷ 462 = 2 remainder 147462 ÷ 147 = 3 remainder 21147 ÷ 21 = 7 remainder 0So, \( \gcd = 21 \). The correct answer is 21.
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Find the GCD of 462 and 1071 using the Euclidean algorithm.
B · 21
1071 ÷ 462 = 2 remainder 147462 ÷ 147 = 3 remainder 21147 ÷ 21 = 7 remainder 0So, \( \gcd = 21 \). The correct answer is 21.
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If \( \gcd(a,b) = 6 \) and \( \mathrm{lcm}(a,b) = 72 \), and \(a = 18\), what is \(b\)?
A · 24
Using \( a \times b = \gcd(a,b) \times \mathrm{lcm}(a,b) \),\(18 \times b = 6 \times 72 = 432 \Rightarrow b = \frac{432}{18} = 24\).
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If \( \gcd(36, x) = 12 \) and \( \mathrm{lcm}(36, x) = 180 \), find \(x\).
A · 60
Using \( a \times b = \gcd(a,b) \times \mathrm{lcm}(a,b) \),\(36 \times x = 12 \times 180 = 2160 \Rightarrow x = \frac{2160}{36} = 60\).
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If two numbers have a GCD of 8 and an LCM of 96, which of the following could be the pair?
C · (24, 32)
For (16,48), \( \gcd = 16 \) (not 8), so incorrect.For (24,32), \( \gcd = 8 \) and \( \mathrm{lcm} = 96 \), so correct.Hence, (24,32) is the correct pair.
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Two ropes of lengths 84 m and 126 m are to be cut into pieces of equal length without any remainder. What is the maximum length of each piece?
C · 42 m
The maximum length is the GCD of 84 and 126.\( \gcd(84,126) = 42 \). The correct answer is 42 m.
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A school has 360 students and wants to arrange them in rows such that each row has the same number of students and the number of rows is also the same. What is the greatest possible number of students in each row?
C · 18
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Two ropes of lengths 84 m and 126 m are to be cut into pieces of equal length without any remainder. What is the maximum length of each piece?
C · 42 m
The maximum length is the GCD of 84 and 126.\( \gcd(84,126) = 42 \).
