Quick recall · 275 cards
Short MCQ-style retrieval prompts. Tap a card to reveal the answer.
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What is the next natural number after 8?
C · 9
PYQ
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What is the sum of the first 10 natural numbers?
A · 55
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What is the product of the first 5 natural numbers?
A · 120
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Is the number zero a natural number? a) True b) False c) Cannot be determined d) None of the above
B · False
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Which of the following are whole numbers? −3, 0, 2.5, 4, 5, −1.2
B · 0, 4, 5
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Which of the following has the greatest absolute value? A) –8 B) –4 C) +2 D) 0
A · –8
Absolute value measures distance from zero on number line, ignoring sign.|–8| = 8, |–4| = 4, |+2| = 2, |0| = 0.8 is greatest, so option A (–8) has greatest absolute value.
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Is \( \frac{5}{8}/0 \) a rational number?
A) Yes B) No
B · No
A rational number is of form \( \frac{a}{b} \) where b ≠ 0. Here denominator is 0, so not rational. Division by zero undefined.
PYQ · 2024
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The smallest irrational number by which \( \sqrt{20} \) should be multiplied so as to get a rational number, is:
C · 5
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Find the mean (correct to two decimal places) of first 7 odd prime numbers.
D · 11.29
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How many prime numbers are there between 20 and 50?
D · 7
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Which number is divisible by 2?
C · 46
PYQ
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If the number 517*324 is completely divisible by 3, then the smallest whole number in the place of * will be?
A · 2
PYQ
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The number 21A35B4 is divisible by 3, where A and B are non-zero digits. Then what is the maximum possible value for A+B?
B · 15
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Which of the following is NOT a natural number?
A · 0
Natural numbers start from 1 and go upwards. Zero is not considered a natural number.
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Which property of natural numbers states that the sum of any two natural numbers is also a natural number?
A · Closure Property
Closure property means that the operation on natural numbers results in a natural number.
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Which of the following statements is TRUE about natural numbers?
C · Natural numbers are positive integers starting from 1
Natural numbers are positive integers starting from 1, excluding zero and fractions.
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What is the successor of 99?
B · 100
The successor of a natural number is the next natural number, so successor of 99 is 100.
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What is the predecessor of 1 in natural numbers?
D · No predecessor
1 is the smallest natural number and has no predecessor in natural numbers.
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If the predecessor of a natural number is 49, what is the number?
B · 50
The number whose predecessor is 49 is 50.
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Which point on the number line represents the natural number 7?
A · The 7th point from zero to the right
Natural numbers are represented as points starting from 1 to the right of zero on the number line.
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Which of the following correctly shows the natural numbers on a number line?
A · Points starting at 1 and moving rightwards
Natural numbers start from 1 and are represented as points moving rightwards on the number line.
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If the natural number 5 is represented on the number line, which number is three units to the left of it?
A · 2
Three units left of 5 is 5 - 3 = 2 on the number line.
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What is the result of \( 7 + 5 \) in natural numbers?
A · 12
Sum of 7 and 5 is 12, which is a natural number.
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Calculate \( 15 - 8 \) using natural numbers.
A · 7
Subtracting 8 from 15 gives 7, which is a natural number.
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What is the product of \( 4 \times 6 \) in natural numbers?
A · 24
Multiplying 4 and 6 gives 24, a natural number.
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Find the value of \( (8 + 5) \times 3 \).
A · 39
First add: 8 + 5 = 13, then multiply: 13 \times 3 = 39.
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If \( 20 - x = 7 \), what is the value of \( x \)?
A · 13
Rearranging: \( x = 20 - 7 = 13 \).
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Evaluate \( 5 \times (3 + 4) - 6 \).
A · 29
Calculate inside parentheses first: 3 + 4 = 7, then multiply: 5 \times 7 = 35, finally subtract: 35 - 6 = 29.
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What is the sum of the first 7 natural numbers?
A · 28
Sum of first n natural numbers is \( \frac{n(n+1)}{2} \). For n=7, sum = \( \frac{7 \times 8}{2} = 28 \).
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Find the sum of the first 15 natural numbers.
A · 120
Sum = \( \frac{15 \times 16}{2} = 120 \). Correct answer is 120.
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What is the sum of the first 20 natural numbers?
A · 210
Sum = \( \frac{20 \times 21}{2} = 210 \).
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Find the sum of the first 10 natural numbers using the formula.
A · 55
Sum = \( \frac{10 \times 11}{2} = 55 \).
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What is \( 5! \) (5 factorial)?
A · 120
5! = 5 \times 4 \times 3 \times 2 \times 1 = 120.
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Calculate \( 4! \).
A · 24
4! = 4 \times 3 \times 2 \times 1 = 24.
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What is the value of \( 6! \)?
A · 720
6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720.
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Find the value of \( 7! \).
A · 5040
7! = 7 \times 6! = 7 \times 720 = 5040.
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Which of the following numbers is divisible by 3?
B · 21
21 is divisible by 3 because 21 \div 3 = 7.
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Which of the following is a factor of 36?
B · 9
9 is a factor of 36 because 36 \div 9 = 4.
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Which of the following numbers is NOT divisible by 2?
B · 21
21 is odd and not divisible by 2.
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Find the greatest common factor (GCF) of 12 and 18.
A · 6
Factors of 12: 1,2,3,4,6,12; Factors of 18: 1,2,3,6,9,18; GCF is 6.
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Which of the following is an even natural number?
B · 22
22 is divisible by 2 and hence even.
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Which number is odd?
C · 35
35 is not divisible by 2 and is an odd number.
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Which of the following is an even number?
B · 64
64 is divisible by 2, so it is even.
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Which of these numbers is greater: 123 or 132?
B · 132
132 is greater than 123.
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Arrange the following natural numbers in ascending order: 45, 12, 78.
A · 12, 45, 78
Ascending order means from smallest to largest: 12, 45, 78.
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Which number is the smallest among 150, 105, and 115?
B · 105
105 is the smallest number among the given options.
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A shopkeeper has 12 boxes each containing 15 apples. If he sells 75 apples, how many apples remain?
A · 105
Total apples = 12 \times 15 = 180. After selling 75, remaining = 180 - 75 = 105.
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If a number is multiplied by 4 and then 20 is added, the result is 100. What is the number?
A · 20
Let the number be x. Then 4x + 20 = 100 \Rightarrow 4x = 80 \Rightarrow x = 20.
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A farmer has 48 apples and wants to pack them equally into baskets. If each basket holds 6 apples, how many baskets does he need?
C · 8
Number of baskets = 48 \div 6 = 8.
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If the sum of three consecutive natural numbers is 72, what is the smallest number?
A · 23
Let the numbers be n, n+1, n+2. Sum = 3n + 3 = 72 \Rightarrow 3n = 69 \Rightarrow n = 23. But sum is 72, so smallest is 23.
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Let \(N\) be the smallest natural number such that \(N\) is divisible by 18, 24, and 30, and the sum of its digits is equal to the number of distinct prime factors of \(N\). What is the value of \(N\)?
C · 180
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Consider a natural number \(M\) such that when divided by 7, 11, and 13, the remainders are 3, 6, and 9 respectively. If \(M\) is less than 1000, what is the sum of the digits of \(M\)?
D · 12
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If \(P\) is the product of all natural numbers less than 50 that are coprime to 50, what is the remainder when \(P\) is divided by 50?
B · 49
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Find the number of natural numbers \(n < 10^5\) such that \(n\) is divisible by 3, the sum of digits of \(n\) is divisible by 9, and \(n\) is not divisible by 9.
A · 8888
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Let \(x\) be the smallest natural number such that \(x\) leaves a remainder 2 when divided by 5, remainder 3 when divided by 7, and remainder 4 when divided by 9. What is the sum of the digits of \(x\)?
B · 14
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Let \(S\) be the set of all natural numbers less than 200 that are divisible by 6 but not by 9. What is the sum of all elements in \(S\)?
D · 5940
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Consider the natural number \(N\) such that \(N\) is the product of all natural numbers less than 20 that are relatively prime to 20. Find the remainder when \(N\) is divided by 20.
B · 19
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If \(a\) and \(b\) are natural numbers such that \(a^2 - b^2 = 2019\), and both \(a + b\) and \(a - b\) are natural numbers, how many such pairs \((a,b)\) exist?
A · 4
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Which of the following natural numbers \(n\) satisfies the property that \(n\) divides \(2^n - 2\) but does not divide \(2^{n} - 1\)?
A · 341
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Find the number of natural numbers \(n\) less than 500 such that \(n\) is divisible by 4 or 6 but not by 12.
C · 251
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Find the smallest natural number \(k\) such that \(k!\) (factorial of \(k\)) is divisible by \(2^{10} \times 3^{5} \times 5^{3}\).
A · 25
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Let \(N\) be a natural number such that \(N\) is divisible by 15 and the sum of its digits is 15. If \(N < 500\), how many such numbers exist?
B · 4
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Match the following natural numbers with their number of distinct prime factors:
Column A:
1. 210
2. 231
3. 300
4. 385
Column B:
A. 3
B. 4
C. 2
D. 5
B · 1-B, 2-A, 3-A, 4-C
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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. -7 is an integer.
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Which property of integers states that \( a + b = b + a \) for any integers \( a \) and \( b \)?
C · Commutative Property
The Commutative Property states that the order of addition or multiplication does not affect the result.
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Which of the following is NOT a property of integers?
B · Closure under division
Integers are not closed under division because dividing two integers may result in a non-integer.
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Which of the following integers is positive?
C · 7
Positive integers are greater than zero. 7 is positive.
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Which integer represents zero?
B · 0
Zero is the integer that is neither positive nor negative.
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Which of the following integers is negative?
B · -3
Negative integers are less than zero. -3 is negative.
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Refer to the diagram below of a number line. What is the integer located exactly between -3 and -1?
A · -2
The integer between -3 and -1 on the number line is -2.
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Refer to the number line diagram below. Which integer is 4 units to the right of -5?
A · -1
Moving 4 units to the right of -5 on the number line lands at -1.
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Refer to the number line below. What is the distance between -7 and 3?
B · 10
The distance between -7 and 3 is \( |3 - (-7)| = 10 \).
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Calculate \( (-5) + 8 \).
A · 3
Adding -5 and 8 results in 3.
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What is the value of \( 12 - (-7) \)?
B · 19
Subtracting a negative is equivalent to addition: \( 12 - (-7) = 12 + 7 = 19 \).
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Calculate \( (-4) \times (-6) \).
B · 24
The product of two negative integers is positive: \( (-4) \times (-6) = 24 \).
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What is \( \frac{-36}{6} \)?
B · -6
Dividing -36 by 6 gives -6.
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Calculate \( (-7) + (-9) \).
A · -16
Sum of two negative integers is negative: \( -7 + (-9) = -16 \).
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Simplify \( (-15) - 20 \).
B · -35
Subtracting 20 from -15 results in \( -15 - 20 = -35 \).
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Calculate \( (-3) \times 7 \).
A · -21
Multiplying a negative and positive integer results in a negative integer: \( -3 \times 7 = -21 \).
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Evaluate \( \frac{(-48)}{-8} \).
A · 6
Dividing two negative integers results in a positive integer: \( \frac{-48}{-8} = 6 \).
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Calculate \( (-5) + 3 \times 4 \).
A · 7
According to order of operations, multiply first: \( 3 \times 4 = 12 \), then add: \( -5 + 12 = 7 \).
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Which property is illustrated by \( 3 + (4 + 5) = (3 + 4) + 5 \)?
B · Associative Property
The Associative Property states that grouping of addition does not affect the sum.
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Which of the following demonstrates the distributive property?
C · \( 4 \times (3 + 5) = 4 \times 3 + 4 \times 5 \)
The distributive property states \( a(b + c) = ab + ac \).
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If \( a = -2, b = 3, c = 4 \), which expression shows the associative property of multiplication?
A · \( a \times (b \times c) = (a \times b) \times c \)
Associative property of multiplication states \( a \times (b \times c) = (a \times b) \times c \).
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Refer to the diagram illustrating \( 3 \times (4 + 2) = 3 \times 4 + 3 \times 2 \). Which property is demonstrated?
C · Distributive
This is the distributive property where multiplication distributes over addition.
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What is the absolute value of \( -15 \)?
B · 15
Absolute value is the distance from zero, so \( |-15| = 15 \).
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Which of the following represents \( |x| = 7 \)?
C · \( x = 7 \) or \( x = -7 \)
Absolute value equation \( |x| = 7 \) means \( x = 7 \) or \( x = -7 \).
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If \( |a| = 12 \) and \( a < 0 \), what is the value of \( a \)?
B · -12
Since \( a < 0 \), \( a = -12 \).
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Refer to the number line below. What is the absolute value of the integer marked at -8?
A · 8
Absolute value is the distance from zero, so \( |-8| = 8 \).
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Which integer is greater: -3 or -7?
A · -3
On the number line, -3 is to the right of -7, so -3 is greater.
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Arrange the following integers in ascending order: \( -2, 5, 0, -7 \).
A · -7, -2, 0, 5
Ascending order is from smallest to largest: -7, -2, 0, 5.
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Which integer is smallest among \( -1, -5, 3, 0 \)?
B · -5
-5 is the smallest integer among the options.
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A submarine is at 50 meters below sea level. It rises 30 meters and then sinks 20 meters. What is its final position relative to sea level?
B · -40 meters
Starting at -50, rising 30: -50 + 30 = -20, sinking 20: -20 - 20 = -40 meters.
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A hiker descends 15 meters and then ascends 25 meters. What is the net change in altitude?
A · 10 meters up
Net change: \( -15 + 25 = 10 \) meters up.
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A bank account has a balance of \( -200 \) dollars (overdraft). If \( 350 \) dollars are deposited, what is the new balance?
A · 150
New balance = \( -200 + 350 = 150 \) dollars.
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A temperature was \( -10^\circ C \) in the morning. It dropped by \( 15^\circ C \) by night. What was the temperature at night?
A · -25\(^\circ C\)
Temperature at night = \( -10 - 15 = -25 \) degrees Celsius.
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An elevator is at the 3rd floor. It goes down 7 floors and then up 4 floors. What floor is it on now?
A · 0
Current floor = 3 - 7 + 4 = 0, but since floors below ground are negative, 0 is ground floor, so answer is 0.
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Which of the following integers is divisible by 4?
B · 20
20 is divisible by 4 since \( 20 \div 4 = 5 \) with no remainder.
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Which of the following is a factor of 36?
B · 9
9 is a factor of 36 because \( 36 \div 9 = 4 \).
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Which integer is divisible by both 2 and 3?
B · 18
18 is divisible by both 2 and 3.
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What is the greatest common divisor (GCD) of 24 and 36?
B · 12
The GCD of 24 and 36 is 12.
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Which of the following integers is a prime number?
B · 17
17 is a prime number as it has only two factors: 1 and 17.
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Which integer is composite?
C · 21
21 is composite because it has factors other than 1 and itself (3 and 7).
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Which of the following is NOT a prime number?
C · 9
9 is not prime because it has factors 1, 3, and 9.
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Refer to the sequence in the diagram below: 2, 4, 8, 16, ... What is the next integer in the sequence?
C · 32
Each term doubles the previous term, so next is \( 16 \times 2 = 32 \).
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What is the 6th term of the sequence: 5, 10, 15, 20, ...?
B · 30
This is an arithmetic sequence with common difference 5. \( 6^{th} = 5 + 5 \times (6-1) = 30 \).
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Find the next number in the sequence: -3, -6, -9, -12, ...
B · -15
Sequence decreases by 3 each time, so next is \( -12 - 3 = -15 \).
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Refer to the pattern sequence diagram below: 1, 4, 9, 16, 25, ... What is the 7th term?
C · 49
This sequence is squares of natural numbers: \( n^2 \). The 7th term is \( 7^2 = 49 \).
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Which of the following is NOT an integer?
C · 3.14
Integers include all whole numbers and their negatives, including zero. 3.14 is a decimal and hence not an integer.
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Which of the following properties is true for all integers under addition?
A · Closure property
The set of integers is closed under addition, meaning the sum of any two integers is an integer. Division and multiplicative inverse are not always defined for integers.
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If \( a \) and \( b \) are integers such that \( a + b = 0 \), which of the following must be true?
B · \( a = -b \)
If \( a + b = 0 \), then \( a = -b \), meaning \( a \) is the additive inverse of \( b \).
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Refer to the diagram below showing integers on a number line. Which integer is located exactly 3 units to the left of 2?
A · -1
Moving 3 units left from 2 on the number line leads to \( 2 - 3 = -1 \).
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On the number line shown below, which integer lies exactly halfway between \(-4\) and \(2\)?
A · -1
The midpoint between \(-4\) and \(2\) is \( \frac{-4 + 2}{2} = -1 \).
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Which of the following is the result of \( (-7) + 12 \)?
A · 5
Adding \( -7 \) and \( 12 \) gives \( 12 - 7 = 5 \).
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Calculate \( (-8) \times (-3) \).
B · 24
The product of two negative integers is positive, so \( (-8) \times (-3) = 24 \).
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What is the value of \( \frac{-36}{6} \)?
B · -6
Dividing \( -36 \) by \( 6 \) gives \( -6 \).
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Evaluate \( (-5) - (-9) \).
A · 4
Subtracting a negative is equivalent to addition: \( -5 + 9 = 4 \).
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Refer to the diagram below showing integer operations on a number line. What is the result of moving 4 units right from \(-3\)?
A · 1
Moving 4 units right from \(-3\) corresponds to \( -3 + 4 = 1 \).
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Which of the following shows the commutative property of multiplication for integers?
C · \( 7 \times (-2) = (-2) \times 7 \)
Commutative property for multiplication states \( a \times b = b \times a \). Option C correctly shows this for integers.
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Which of the following expressions demonstrates the associative property of addition for integers?
A · \( (4 + 5) + (-3) = 4 + (5 + (-3)) \)
Associative property of addition states \( (a + b) + c = a + (b + c) \). Option A correctly illustrates this.
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Identify the expression that correctly applies the distributive law for integers.
A · \( 3 \times (4 + (-2)) = 3 \times 4 + 3 \times (-2) \)
Distributive law states \( a \times (b + c) = a \times b + a \times c \). Option A correctly applies this.
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Refer to the diagram below illustrating the distributive property on a number line. Which of the following is true?
A · \( 2 \times (3 + (-1)) = 2 \times 3 + 2 \times (-1) \)
Distributive property holds as \( 2 \times (3 + (-1)) = 2 \times 3 + 2 \times (-1) \).
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Which integer is greater than \(-3\) but less than \(2\)?
B · 0
0 lies between \(-3\) and \(2\).
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Arrange the following integers in ascending order: \( -2, 5, -7, 0 \).
A · -7, -2, 0, 5
Ascending order means from smallest to largest: \( -7 < -2 < 0 < 5 \).
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Which of the following statements is true?
B · \( -1 < 0 \)
Among integers, \( -1 < 0 \) is true. Negative numbers are less than zero.
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What is the absolute value of \(-15\)?
B · 15
Absolute value of a number is its distance from zero on the number line, always non-negative. So, \( | -15 | = 15 \).
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Which of the following integers has the smallest absolute value?
B · 0
Absolute value of 0 is 0, which is the smallest possible.
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If \( |x| = 7 \), which of the following could be the value of \( x \)?
C · Both 7 and -7
Absolute value of \( x \) is 7 means \( x = 7 \) or \( x = -7 \).
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A submarine is at a depth of 120 meters below sea level. It rises 45 meters. What is its new position relative to sea level?
B · -75 meters
Starting at \(-120\) meters, rising 45 meters means \( -120 + 45 = -75 \) meters below sea level.
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A temperature drops from 5\(^\circ\)C to -3\(^\circ\)C. What is the change in temperature?
A · -8\(^\circ\)C
Change = Final - Initial = \( -3 - 5 = -8 \)\(^\circ\)C, indicating a drop of 8 degrees.
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A hiker descends 50 meters from the top of a hill and then ascends 30 meters. What is the hiker's net change in elevation?
B · -20 meters
Net change = \( -50 + 30 = -20 \) meters, meaning 20 meters below the starting point.
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A bank account has a balance of \( -\$200 \). After a deposit of \( \$350 \), what is the new balance?
A · \$150
New balance = \( -200 + 350 = 150 \) dollars positive.
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If \( x = -3 \) and \( y = 5 \), what is the value of \( 2x - 3y \)?
A · -21
Calculate: \( 2(-3) - 3(5) = -6 - 15 = -21 \). Correction: The correct answer is -21.
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Solve for \( x \): \( x + (-7) = 4 \).
A · 11
Add 7 to both sides: \( x = 4 + 7 = 11 \).
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If \( 3x - 5 = 16 \), what is the value of \( x \)?
A · 7
Add 5 to both sides: \( 3x = 21 \), then \( x = 7 \). Correction: The correct answer is 7.
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Refer to the number line below. If \( x \) is at \(-2\) and \( y \) is at \(3\), what is the value of \( x^2 + y \)?
A · 7
Calculate \( (-2)^2 + 3 = 4 + 3 = 7 \). Correction: The correct answer is 7.
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Let \(a\) and \(b\) be two integers such that \(\gcd(a,b) = 17\) and \(\mathrm{lcm}(a,b) = 17^3 \times 5^2\). If \(a = 17^2 \times 5^x\) and \(b = 17 \times 5^y\) for some integers \(x,y \geq 0\), find \(|x - y|\).
B · 2
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Consider integers \(x,y,z\) satisfying \(x+y+z=0\) and \(\gcd(x,y,z) = 1\). If \(x^2 + y^2 + z^2 = 2023\), what is the value of \(\gcd(x^3 + y^3 + z^3, x^2y + y^2z + z^2x)\)?
A · 1
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If \(m,n\) are integers such that \(m^2 - n^2 = 2021\) and \(\gcd(m,n) = 1\), which of the following can be the value of \(\gcd(m+n, m-n)\)?
B · 2
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Let \(a,b\) be integers such that \(a^2 + b^2 = 2027\) and \(\gcd(a,b) = d\). If \(d > 1\), which of the following must be true?
A · d divides 2027
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For integers \(x,y\), define \(S = \frac{x^3 - y^3}{x - y}\). If \(S\) is divisible by 2021, which of the following must be true?
B · \(x^2 + xy + y^2\) is divisible by 2021
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Let \(a,b,c\) be integers such that \(a+b+c=0\) and \(\gcd(a,b,c) = 1\). If \(a^2 + b^2 + c^2 = 2029\), which of the following is the value of \(\gcd(a+b, b+c, c+a)\)?
B · 2
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If \(x,y\) are integers such that \(x^2 + y^2 = 2025\) and \(\gcd(x,y) = 3\), what is the value of \(\gcd(x+y, x-y)\)?
C · 6
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Let \(a,b\) be integers such that \(a+b = 2023\) and \(\gcd(a,b) = 1\). If \(a^3 + b^3\) is divisible by 2023, what is \(\gcd(a^2 - ab + b^2, 2023)\)?
D · 2023
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Given integers \(x,y\) with \(\gcd(x,y) = 1\) and \(x^2 - xy + y^2 = 2027\), which of the following is true about \(\gcd(x+y, x-y)\)?
A · 1
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If integers \(a,b\) satisfy \(a^2 + b^2 = 2023\) and \(a+b\) divides \(a^3 + b^3\), what is \(\gcd(a+b, a-b)\)?
A · 1
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Let \(a,b\) be integers with \(a+b=2023\) and \(\gcd(a,b) = 1\). If \(a^2 + b^2\) is divisible by 2023, what is \(\gcd(a-b, 2023)\)?
A · 1
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If integers \(x,y\) satisfy \(x^2 - y^2 = 2023\) and \(\gcd(x,y) = 1\), which of the following is possible for \(\gcd(x+y, x-y)\)?
B · 2
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For integers \(a,b\) with \(\gcd(a,b) = 1\), if \(a^2 + b^2 = 2021\), which of the following must be true about \(\gcd(a+b, a-b)\)?
B · 2
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Given integers \(x,y\) such that \(x+y=2023\) and \(x^3 + y^3\) divisible by 2023, what is \(\gcd(x^2 - xy + y^2, 2023)\)?
D · 2023
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If integers \(a,b\) satisfy \(a^2 - b^2 = 2027\) and \(\gcd(a,b) = 1\), what is the possible value of \(\gcd(a+b, a-b)\)?
B · 2
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Let \(x,y\) be integers such that \(x^2 + y^2 = 2029\) and \(\gcd(x,y) = d > 1\). Which of the following must be true?
A · d divides 2029
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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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Identify the prime number from the following set: 4, 6, 9, 11
D · 11
11 is prime as it is divisible only by 1 and itself.
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Which of the following is NOT a prime number?
D · 9
9 is not prime because it is divisible by 3.
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Which of the following numbers is prime?
B · 53
53 is prime; it has no divisors other than 1 and itself.
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Which property is true for all prime numbers greater than 2?
B · They are all odd numbers
All prime numbers greater than 2 are odd because 2 is the only even prime number.
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If \( p \) and \( p+2 \) are both prime numbers, what are they called?
B · Twin primes
Primes that differ by 2 are called twin primes.
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Which of the following statements about prime numbers is FALSE?
C · All prime numbers are odd
Not all prime numbers are odd; 2 is an even prime number.
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Which of the following is a property of prime numbers?
C · A prime number has exactly two distinct positive divisors
By definition, a prime number has exactly two distinct positive divisors: 1 and itself.
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What is the prime factorization of 84?
B · \( 2^2 \times 3 \times 7 \)
84 = 2 \times 2 \times 3 \times 7 = \( 2^2 \times 3 \times 7 \).
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Find the prime factorization of 360.
A · \( 2^3 \times 3^2 \times 5 \)
360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5 = \( 2^3 \times 3^2 \times 5 \).
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Which of the following is the correct prime factorization of 2310?
A · \( 2 \times 3 \times 5 \times 7 \times 11 \)
2310 = 2 \times 3 \times 5 \times 7 \times 11.
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Find the prime factorization of 1001.
A · \( 7 \times 11 \times 13 \)
1001 = 7 \times 11 \times 13.
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Which method is commonly used to test if a number \( n \) is prime?
B · Check divisibility up to \( \sqrt{n} \)
To test primality, it is sufficient to check divisibility up to \( \sqrt{n} \).
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Which of the following numbers is prime based on divisibility tests up to \( \sqrt{n} \)?
A · 29
29 is prime; it is not divisible by any prime less than or equal to \( \sqrt{29} \approx 5.38 \).
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Which shortcut can be used to quickly determine that 91 is not prime?
B · Divisible by 7
91 = 7 \times 13, so it is divisible by 7 and not prime.
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Which of the following numbers is prime using Fermat's little theorem as a primality test candidate?
B · 17
17 is prime; 341, 561, and 91 are Carmichael or composite numbers that can fool Fermat's test.
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Which of the following best describes the distribution of prime numbers?
B · Prime numbers become less frequent as numbers increase
Prime numbers become less frequent as numbers increase, but they never stop appearing.
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Which statement about the distribution of primes is true?
A · There are infinitely many primes
There are infinitely many prime numbers, as proved by Euclid.
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Which of the following is a Mersenne prime?
D · Both A and C
Both 31 and 127 are Mersenne primes because they are of the form \( 2^p - 1 \) where \( p \) is prime.
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Twin primes are pairs of primes that differ by:
B · 2
Twin primes differ by 2, for example (11, 13) or (17, 19).
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Which of the following is NOT a special type of prime?
D · Composite prime
Composite numbers are not prime; 'composite prime' is a contradiction.
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If \( p \) is a prime number, which of the following is always true for \( p eq 2 \)?
B · \( p \) is odd
All prime numbers except 2 are odd.
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Using prime factorization, find the greatest common divisor (GCD) of 48 and 180.
A · 12
Prime factors: 48 = \( 2^4 \times 3 \), 180 = \( 2^2 \times 3^2 \times 5 \). GCD = \( 2^2 \times 3 = 12 \).
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If a number \( n \) is divisible by two distinct primes \( p \) and \( q \), which of the following must be true?
B · \( n \) is composite
If \( n \) has two distinct prime factors, it must be composite.
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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 itself.
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Identify the prime number from the following set: 4, 6, 9, 11
D · 11
11 is prime as it has no divisors other than 1 and 11.
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Which of the following is NOT a prime number?
C · 9
9 is not prime because it is divisible by 3.
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Which number is prime among the following: 51, 53, 55, 57?
B · 53
53 is prime; the others are divisible by numbers other than 1 and themselves.
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Which of the following is a unique property of prime numbers greater than 2?
B · They are odd
All prime numbers greater than 2 are odd because 2 is the only even prime.
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Which of the following statements about prime numbers is TRUE?
B · 2 is the only even prime number
2 is the only even prime number; all others are odd.
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If \( p \) is a prime number greater than 3, which of the following is always true?
B · \( p^2 - 1 \) is divisible by 24
For any prime \( p > 3 \), \( p^2 - 1 = (p-1)(p+1) \) is divisible by 24.
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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 is the prime factorization.
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Find the prime factors of 210.
A · 2, 3, 5, 7
210 = 2 \( \times \) 3 \( \times \) 5 \( \times \) 7.
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The number 462 can be expressed as a product of primes as:
A · 2 \( \times \) 3 \( \times \) 7 \( \times \) 11
462 = 2 \( \times \) 3 \( \times \) 7 \( \times \) 11.
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Which of the following is the prime factorization of 1001?
A · 7 \( \times \) 11 \( \times \) 13
1001 = 7 \( \times \) 11 \( \times \) 13.
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Which test can quickly determine if 29 is prime?
A · Check divisibility up to \( \sqrt{29} \)
To test primality, check divisibility by primes up to \( \sqrt{29} \approx 5.38 \).
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Which of the following is a valid method to test if 91 is prime?
A · Check divisibility by primes less than or equal to 9
Check divisibility by primes \( \leq \sqrt{91} \approx 9.5 \) (2,3,5,7). 91 is divisible by 7.
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Using Fermat's little theorem, which number is likely prime?
C · 13
13 is prime; 561 and 1105 are Carmichael numbers (composite but pass Fermat test for some bases).
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Which of the following numbers passes the Miller-Rabin primality test for base 2 and is prime?
B · 17
17 is prime and passes Miller-Rabin test; others are composite.
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Which statement best describes the distribution of prime numbers?
A · Primes become less frequent as numbers get larger
Prime numbers become less frequent as numbers increase, though infinitely many exist.
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Which of the following approximates the number of primes less than 1000?
A · About 168
There are 168 primes less than 1000 approximately.
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Which theorem describes the asymptotic distribution of prime numbers?
A · Prime Number Theorem
The Prime Number Theorem describes how primes are distributed among natural numbers.
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Which of the following pairs are twin primes?
D · Both A and C
Twin primes are pairs of primes differing by 2; (17,19) and (29,31) qualify.
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Which of the following is a Mersenne prime?
D · 127
127 = 2^7 - 1 is a Mersenne prime.
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Which of the following statements about Mersenne primes is TRUE?
A · They are primes of the form \( 2^p - 1 \) where \( p \) is prime
Mersenne primes have the form \( 2^p - 1 \) where \( p \) itself is prime.
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Which of the following problems can be solved using prime numbers?
A · Finding the greatest common divisor (GCD)
Prime factorization helps in finding the GCD of numbers.
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In cryptography, which property of prime numbers is most useful?
A · Their indivisibility
Indivisibility of primes underpins many cryptographic algorithms.
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If \( n = p \times q \) where \( p \) and \( q \) are distinct large primes, which problem is considered hard and forms the basis of RSA encryption?
A · Factoring \( n \) into \( p \) and \( q \)
Factoring large semiprimes \( n \) is computationally hard, securing RSA encryption.
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Let p and q be distinct odd primes such that p + q = 2023 and p - q divides p^2 - q^2. Which of the following is true about p and q?
D · No such primes p and q exist
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If p is a prime greater than 3, which of the following must be true about p^2 - 1?
A · p^2 - 1 is divisible by 24
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Consider the set of primes p such that p divides 2^p - 2. Which of the following statements is correct?
A · All primes p satisfy p | 2^p - 2
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Let p be a prime such that p divides (n^2 + 1) for some integer n. Which of the following must be true about p?
A · p = 2 or p ≡ 1 (mod 4)
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If p and q are primes such that p divides q^2 + q + 1 and q divides p^2 + p + 1, which of the following pairs (p, q) is possible?
D · No such prime pairs exist
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Which of the following primes p satisfies the condition that p divides 2^{p-1} - 1 but does not divide 2^k - 1 for any k < p - 1?
D · p = 19
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If p is a prime such that p divides the sum of the first p natural numbers, which of the following is true?
B · p divides p(p+1)/2 for all primes p
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Let p be a prime greater than 3. Which of the following statements about the factorial (p-1)! modulo p is true?
A · (p-1)! ≡ -1 (mod p)
Step 1: Wilson's theorem states (p-1)! ≡ -1 (mod p) for prime p.
Step 2: This is true for all primes p.
Step 3: Hence option A is correct.
Step 4: Options B, C, D contradict Wilson's theorem.
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If p and q are primes such that p divides q - 1 and q divides p - 1, which of the following pairs (p, q) is possible?
D · No such pairs exist
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Let p be a prime such that 2p + 1 is also prime. Which of the following is true about p?
A · p must be a Sophie Germain prime
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For a prime p > 3, consider the number S = 1^p + 2^p + 3^p + ... + (p-1)^p. Which of the following is true about S modulo p?
A · S ≡ 0 (mod p)
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If p is a prime greater than 3, which of the following must divide the product (p-2)! + 1?
D · None of the above
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Which of the following primes p satisfies that p divides the number 2^{p-1} + 1?
D · No prime p satisfies this
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Let p be a prime such that p divides the number 10^{p} - 10. Which of the following statements is true?
B · p divides 10^{p-1} - 1
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For a prime p > 3, which of the following is the remainder when (p-1)! + 1 is divided by p^2?
B · p
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If p is a prime and divides the number 3^{p} - 3, which of the following is true?
B · p divides 3^{p-1} - 1
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Which of the following primes p satisfies that p divides the number (p-2)!?
D · No prime p divides (p-2)!
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Assertion (A): For any prime p > 3, p^2 - 1 is divisible by 24.
Reason (R): p^2 - 1 = (p-1)(p+1), and among two consecutive even numbers one is divisible by 4 and one is divisible by 3.
Choose the correct option:
A · Both A and R are true and R is the correct explanation of A
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Which of the following best defines divisibility in number theory?
B · A number \(a\) is divisible by \(b\) if \(a \div b\) leaves no remainder
Divisibility means that when \(a\) is divided by \(b\), the remainder is zero.
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If \(a\) is divisible by \(b\), which of the following must be true?
B · \(a = b \times k\) for some integer \(k\)
Divisibility means \(a\) can be expressed as \(b\) times an integer \(k\), i.e., \(a = b \times k\).
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Which of the following statements is true about divisibility?
A · If \(a\) divides \(b\) and \(b\) divides \(c\), then \(a\) divides \(c\)
Divisibility is transitive: if \(a\) divides \(b\) and \(b\) divides \(c\), then \(a\) divides \(c\).
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Which of the following numbers is divisible by 2?
B · 2468
A number is divisible by 2 if its last digit is even. 2468 ends with 8, which is even.
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Which of the following numbers is divisible by 5?
A · 1230
A number is divisible by 5 if it ends with 0 or 5. 1230 ends with 0.
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Which of the following numbers is divisible by 10?
A · 3450
A number is divisible by 10 if it ends with 0. 3450 ends with 0.
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Which of the following numbers is divisible by 3?
A · 123
A number is divisible by 3 if the sum of its digits is divisible by 3. Sum of digits of 123 is 6, which is divisible by 3.
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Determine if 462 is divisible by 6.
A · Yes, because it is divisible by both 2 and 3
A number is divisible by 6 if it is divisible by both 2 and 3. 462 is even and sum of digits (4+6+2=12) is divisible by 3.
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Is 121 divisible by 11?
A · Yes, because the difference between the sum of digits in odd and even places is 0
For divisibility by 11, the difference between the sum of digits in odd and even positions must be 0 or multiple of 11. For 121, (1+1) - (2) = 0.
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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 18, which is divisible by 9.
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Is 123456 divisible by 4?
A · Yes, because last two digits 56 are divisible by 4
A number is divisible by 4 if the last two digits form a number divisible by 4. 56 is divisible by 4.
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Which of the following numbers is divisible by 8?
B · 123448
A number is divisible by 8 if the last three digits form a number divisible by 8. 448 is divisible by 8.
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Find the smallest number divisible by both 3 and 11.
A · 33
The smallest number divisible by both 3 and 11 is their least common multiple (LCM), which is 33.
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If a number is divisible by 2 and 9, which of the following must it be divisible by?
A · 18
If a number is divisible by 2 and 9, it must be divisible by their LCM, which is 18.
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Which of the following numbers is divisible by 7?
A · 203
203 is divisible by 7 because \(7 \times 29 = 203\).
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Which of the following is a divisibility test for 13?
A · Multiply the last digit by 4 and add to the rest; if result divisible by 13, original number is divisible
For 13, multiply the last digit by 4 and add to the rest; if the result is divisible by 13, the number is divisible by 13.
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Is 221 divisible by 13?
A · Yes, because applying the test yields a multiple of 13
Using the test: last digit 1 \(\times 4 = 4\), add to 22 \(= 26\), which is divisible by 13, so 221 is divisible by 13.
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Which of the following numbers is divisible by 17?
D · 272
272 is divisible by 17 because \(17 \times 16 = 272\).
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Which property of divisibility states that if \(a\) divides \(b\) and \(a\) divides \(c\), then \(a\) divides \(b + c\)?
B · Closure Property
Closure property of divisibility states that the sum of two numbers divisible by \(a\) is also divisible by \(a\).
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If \(a\) divides \(b\) and \(b\) divides \(c\), then which property ensures \(a\) divides \(c\)?
B · Transitive Property
Transitive property of divisibility states that if \(a\) divides \(b\) and \(b\) divides \(c\), then \(a\) divides \(c\).
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Which of the following is NOT true about divisibility properties?
B · Divisibility is symmetric
Divisibility is not symmetric; if \(a\) divides \(b\), \(b\) does not necessarily divide \(a\).
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Find the greatest common divisor (GCD) of 48 and 60.
A · 12
The GCD of 48 and 60 is 12, which is the largest number dividing both.
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What is the least common multiple (LCM) of 6 and 8?
A · 24
LCM of 6 and 8 is 24, the smallest number divisible by both.
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If the GCD of two numbers is 5 and their LCM is 60, what is the product of the two numbers?
A · 300
Product of two numbers = GCD \(\times\) LCM = 5 \(\times\) 60 = 300.
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Which of the following numbers is a common divisor of 36 and 48?
D · 12
12 divides both 36 and 48, making it a common divisor.
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Simplify the fraction \( \frac{84}{126} \) using divisibility rules.
A · \( \frac{2}{3} \)
GCD of 84 and 126 is 42. Dividing numerator and denominator by 42 gives \( \frac{2}{3} \).
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Factorize 90 using divisibility rules.
A · \( 2 \times 3^2 \times 5 \)
90 = 2 \(\times\) 3 \(\times\) 3 \(\times\) 5 = \( 2 \times 3^2 \times 5 \).
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Which of the following fractions is in simplest form?
D · \( \frac{13}{27} \)
\( \frac{13}{27} \) is in simplest form because 13 and 27 have no common divisors other than 1.
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A number is divisible by 7 if you double the last digit and subtract it from the rest of the number and the result is divisible by 7. Using this rule, is 203 divisible by 7?
A · Yes, because 20 - 2 \(\times\) 3 = 14 is divisible by 7
Double last digit 3 \(\times 2 = 6\), subtract from 20: 20 - 6 = 14, which is divisible by 7, so 203 is divisible by 7.
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Which of the following best defines divisibility of integers?
B · An integer \(a\) is divisible by \(b\) if \(a\) divided by \(b\) leaves a remainder zero
Divisibility means that when one integer \(a\) is divided by another integer \(b\), the remainder is zero.
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If \(x\) and \(y\) are integers and \(x\) is divisible by \(y\), which of the following must be true?
B · \(x = y \times k\) for some integer \(k\)
If \(x\) is divisible by \(y\), then \(x\) can be expressed as \(y\) multiplied by some integer \(k\).
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Which of the following statements is NOT true about divisibility?
C · If \(a\) divides \(b\), then \(b\) divides \(a\)
Divisibility is not symmetric; if \(a\) divides \(b\), it does not imply \(b\) divides \(a\).
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Which of the following numbers is divisible by 2?
B · 2468
A number is divisible by 2 if its last digit is even. 2468 ends with 8, which is even.
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Which of the following numbers is divisible by 9?
B · 987654
A number is divisible by 9 if the sum of its digits is divisible by 9. Sum of digits of 987654 is 9+8+7+6+5+4=39, and 39 is divisible by 9.
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Which of the following numbers is divisible by 5?
B · 1230
A number is divisible by 5 if its last digit is 0 or 5. 1230 ends with 0.
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Which of the following numbers is divisible by 6?
A · 1236
A number is divisible by 6 if it is divisible by both 2 and 3. 1236 is even and sum of digits (1+2+3+6=12) is divisible by 3.
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Determine which number is divisible by 8:
D · 12368
A number is divisible by 8 if the last three digits form a number divisible by 8. 368 is divisible by 8 (8×46=368).
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Which of the following numbers is divisible by 4 but NOT by 8?
A · 1324
1324 ends with 24 which is divisible by 4 but not by 8, so 1324 is divisible by 4 but not by 8.
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Find the smallest 4-digit number divisible by both 3 and 9 but not by 6.
B · 1017
Divisible by 9 means divisible by 3. Not divisible by 6 means not divisible by 2 (not even). 1017 sum of digits is 9, divisible by 9, and 1017 is odd.
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Which of the following numbers is divisible by 11?
A · 2728
A number is divisible by 11 if the difference between the sum of digits in odd and even positions is a multiple of 11 (including 0). For 2728: (2+2) - (7+8) = 4 - 15 = -11, divisible by 11.
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A number is divisible by 12 if it is divisible by which pair of numbers?
A · 3 and 4
12 = 3 × 4, so a number divisible by both 3 and 4 is divisible by 12.
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Which of the following numbers is divisible by 15?
A · 1230
A number is divisible by 15 if it is divisible by both 3 and 5. 1230 ends with 0 (divisible by 5) and sum of digits is 6 (divisible by 3).
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Which of the following numbers is divisible by 25?
B · 10250
A number is divisible by 25 if its last two digits are 00, 25, 50, or 75. 10250 ends with 50.
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Find the smallest 5-digit number divisible by 11 and 25.
A · 11000
LCM of 11 and 25 is 275. The smallest 5-digit multiple of 275 is 11000 (275 × 40).
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If a number is divisible by both 4 and 15, which of the following must be true?
A · It is divisible by 60
LCM of 4 and 15 is 60, so the number must be divisible by 60.
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Which of the following numbers is a factor of 360 based on divisibility rules?
D · 30
360 is divisible by 30 because 360 ÷ 30 = 12 (an integer). 30 is a factor of 360.
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If a number is divisible by 9 and 12, which of the following numbers must it be divisible by?
B · 36
LCM of 9 and 12 is 36, so the number must be divisible by 36.
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A number \(n\) is divisible by 4 and 25 but not by 100. Which of the following could be \(n\)?
B · 300
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Using divisibility rules, determine if 123456789 is divisible by 3 without performing division.
A · Yes, because sum of digits is divisible by 3
A number is divisible by 3 if the sum of its digits is divisible by 3. Sum of digits is 45, which is divisible by 3.
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Which of the following numbers is divisible by 18 without performing division?
D · 288
A number is divisible by 18 if it is divisible by both 9 and 2. 288 is even and sum of digits (2+8+8=18) is divisible by 9.
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Without actual division, determine which number is divisible by 24:
A · 3456
24 = 3 × 8. 3456 is divisible by 3 (sum of digits 18 divisible by 3) and by 8 (last three digits 456 divisible by 8).
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Which of the following is a common misconception about divisibility by 3?
B · A number ending with 3 is divisible by 3
Ending with 3 does not guarantee divisibility by 3. Divisibility by 3 depends on the sum of digits, not the last digit.
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Which of the following statements about divisibility is false?
C · All even numbers are divisible by 3
Not all even numbers are divisible by 3. For example, 4 is even but not divisible by 3.
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Which of the following numbers is prime based on divisibility rules?
A · 29
29 is not divisible by 2, 3, 5 or any smaller prime, so it is prime.
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Which of the following statements correctly describes the relationship between divisibility and prime numbers?
B · Prime numbers have exactly two distinct positive divisors: 1 and itself
Prime numbers have exactly two distinct positive divisors: 1 and the number itself.
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If a number is divisible by 7 and 13, which of the following must be true?
B · It is divisible by 91
LCM of 7 and 13 is 91, so the number must be divisible by 91.
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Which of the following composite numbers is divisible by exactly three distinct prime numbers?
A · 30
30 = 2 × 3 × 5, divisible by exactly three distinct primes.
