How is a factor different from a multiple?

A factor divides into a number, while a multiple is the result of multiplying a number by a whole number. For example, 4 is a factor of 20, but 20 is a multiple of 4. Factors are smaller than or equal to the number; multiples are equal to or larger than it.

Factor — Definition, Formula & Examples

A factor is a whole number that divides evenly into another number with no remainder. For example, 3 is a factor of 12 because 12 ÷ 3 = 4 with nothing left over.

An integer $a$ is a factor of an integer $b$ if there exists an integer $k$ such that $b = a \times k$ , meaning $a$ divides $b$ exactly with a remainder of zero.

Key Formula

b = a \times k

Where:

$a$ = A factor of b
$b$ = The number being divided
$k$ = The other factor that pairs with a

How It Works

To find all the factors of a number, test each whole number starting from 1 up to that number and check whether it divides in evenly. A useful shortcut is to stop testing once you pass the square root of the number, because factors come in pairs. For instance, if you find that 2 is a factor of 12, you automatically know 6 is also a factor because

2 \times 6 = 12

. Every whole number greater than 0 has at least two factors: 1 and itself.

Worked Example

Problem: Find all the factors of 18.

Step 1: Start with 1. Since 18 ÷ 1 = 18, both 1 and 18 are factors.

1 \times 18 = 18

Step 2: Try 2. Since 18 ÷ 2 = 9, both 2 and 9 are factors.

2 \times 9 = 18

Step 3: Try 3. Since 18 ÷ 3 = 6, both 3 and 6 are factors.

3 \times 6 = 18

Step 4: Try 4. Since 18 ÷ 4 = 4.5, the division is not even, so 4 is not a factor. Try 5: 18 ÷ 5 = 3.6, so 5 is not a factor either. Since we have passed the square root of 18 (about 4.24), we can stop.

Answer: The factors of 18 are 1, 2, 3, 6, 9, and 18.

Another Example

Problem: Is 7 a factor of 35?

Step 1: Divide 35 by 7.

35 \div 7 = 5

Step 2: The result is a whole number with no remainder, so 7 divides 35 evenly.

Answer: Yes, 7 is a factor of 35 because

7 \times 5 = 35

Visualization

Why It Matters

Finding factors is a key skill in elementary and middle school math, especially when simplifying fractions or finding the greatest common factor of two numbers. It also lays the groundwork for prime factorization and understanding divisibility rules, which appear throughout pre-algebra and algebra courses.

Common Mistakes

Mistake: Forgetting that 1 and the number itself are always factors.

Correction: Every whole number greater than 0 is divisible by 1 and by itself. Always include both when listing factors.

Mistake: Confusing factors with multiples.

Correction: Factors divide into a number (they are equal to or smaller). Multiples result from multiplying (they are equal to or larger). For example, factors of 10 include 2 and 5, while multiples of 10 include 10, 20, 30, and so on.