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Every number has certain numbers that divide it exactly without leaving any remainder. These numbers are called factors. For example, factors of 6 are 1, 2, 3, and 6 because each divides 6 exactly.
Understanding factors helps us classify numbers into two special groups: prime numbers and composite numbers. This classification is important because it helps us solve many math problems, especially in competitive exams.
Before we start, it is important to know that the number 1 is neither prime nor composite, and 0 is a special number that does not fit into these categories.
A factor of a number is a number that divides it exactly without leaving a remainder.
Example: Factors of 10 are 1, 2, 5, and 10 because:
A prime number is a number greater than 1 that has exactly two distinct factors: 1 and the number itself.
In other words, a prime number cannot be divided evenly by any other number except 1 and itself.
Examples of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...
Notice that 2 is the only even prime number because every other even number can be divided by 2, making them composite.
A composite number is a number greater than 1 that has more than two factors.
This means composite numbers can be divided evenly by numbers other than 1 and itself.
Examples of composite numbers: 4, 6, 8, 9, 12, 15, 18, 20, 24, 30, ...
We can find the prime factors of composite numbers by breaking them down step-by-step using a factor tree.
From the factor tree above, the prime factors of 12 are 2, 2, and 3.
Number 1: It has only one factor, which is 1 itself. Since prime numbers need exactly two factors, 1 is neither prime nor composite.
Number 0: It is a special number that can be divided by every number, so it does not fit into prime or composite categories.
Remember: 1 is neither prime nor composite. 0 is not classified as either.
Divisibility rules help us quickly check if a number can be divided by another number without performing full division. This is very useful to identify factors and decide if a number is prime or composite.
| Divisor | Rule | Example |
|---|---|---|
| 2 | Number ends with 0, 2, 4, 6, or 8 | 24 ends with 4 -> divisible by 2 |
| 3 | Sum of digits is divisible by 3 | 123 -> 1+2+3=6, 6 divisible by 3 -> 123 divisible by 3 |
| 5 | Number ends with 0 or 5 | 45 ends with 5 -> divisible by 5 |
Step 1: List factors of 17 by checking divisibility from 1 to 17.
Step 2: 17 / 1 = 17 (no remainder), 17 / 17 = 1 (no remainder).
Check other numbers: 2, 3, 4 ... none divide 17 exactly.
Step 3: Since only 1 and 17 divide 17 exactly, it has exactly two factors.
Answer: 17 is a prime number.
Step 1: Start with 18 at the top.
Step 2: Find two factors of 18: 2 and 9 (because 2 x 9 = 18).
Step 3: Break down 9 into factors: 3 and 3.
Step 4: All factors at the ends (2, 3, 3) are prime numbers.
Answer: 18 is composite because it has more than two factors (1, 2, 3, 6, 9, 18).
Step 1: Check divisibility by 2: 35 ends with 5, so not divisible by 2.
Step 2: Check divisibility by 3: sum of digits 3 + 5 = 8, which is not divisible by 3.
Step 3: Check divisibility by 5: number ends with 5, so divisible by 5.
Step 4: Since 35 is divisible by 5 (other than 1 and 35), it has more than two factors.
Answer: 35 is a composite number.
Step 1: List numbers: 21, 22, 23, 24, 25, 26, 27, 28, 29.
Step 2: Check each number:
Answer: The prime numbers between 20 and 30 are 23 and 29.
Step 1: Start with 60.
Step 2: Break 60 into two factors: 6 and 10 (6 x 10 = 60).
Step 3: Break 6 into 2 and 3 (both prime).
Step 4: Break 10 into 2 and 5 (both prime).
Step 5: Collect all prime factors: 2, 2, 3, and 5.
Answer: Prime factorization of 60 is \(2 \times 2 \times 3 \times 5\) or \(2^2 \times 3 \times 5\).
When to use: When quickly checking if an even number is prime.
When to use: When testing if a number is composite without full factorization.
When to use: For checking primality of larger numbers efficiently.
When to use: When performing prime factorization.
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