How to Find the Factors of a Number

Factoring a number means finding all the whole numbers that divide into it evenly. It's one of those fundamental skills that keeps showing up throughout math — in fractions, algebra, and beyond.

What Is a Factor?

A factor of a number is any whole number that divides into it evenly — with no remainder.

For example, 3 is a factor of 12, because $12 \div 3 = 4$ exactly. But 5 is not a factor of 12, because $12 \div 5 = 2.4$ — there's a remainder.

A few rules that are always true:

1 is a factor of every number. (Every number divides by 1 evenly.)
Every number is a factor of itself. (Any number divided by itself equals 1.)
So every number has at least two factors: 1 and itself.

Finding All Factors with Factor Pairs

The most reliable way to find all the factors of a number is to look for factor pairs — two numbers that multiply together to give you the original number. If you work your way up systematically from 1, you won't miss any.

Example: Find all factors of 36

Start with 1 and work up:

$1 \times 36 = 36$ → factors: 1, 36
$2 \times 18 = 36$ → factors: 2, 18
$3 \times 12 = 36$ → factors: 3, 12
$4 \times 9 = 36$ → factors: 4, 9
$5 \times ? = 36$ → 36 ÷ 5 = 7.2, not a whole number. Skip.
$6 \times 6 = 36$ → factor: 6

Once the two numbers in the pair meet in the middle (both equal 6), you've found them all.

All factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Example: Find all factors of 42

$1 \times 42 = 42$ → 1, 42
$2 \times 21 = 42$ → 2, 21
$3 \times 14 = 42$ → 3, 14
$4 \times ? = 42$ → 42 ÷ 4 = 10.5. Skip.
$5 \times ? = 42$ → 42 ÷ 5 = 8.4. Skip.
$6 \times 7 = 42$ → 6, 7

All factors of 42: 1, 2, 3, 6, 7, 14, 21, 42

Tip: You can stop checking once your test number is larger than the square root of the original number. At that point, any remaining pairs have already been found from the other direction.

Prime Numbers

A prime number is a number that has exactly two factors: 1 and itself. It can't be broken down further.

The first several primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...

A number with more than two factors is called composite. For example, 12 is composite because it has factors 1, 2, 3, 4, 6, and 12.

The number 1 is neither prime nor composite — it's a special case.

Is 37 prime? Check by dividing by each prime up to $\sqrt{37} \approx 6.1$: try 2, 3, 5. None divide evenly. So yes, 37 is prime.

Prime Factorization

Every composite number can be broken down into a product of prime numbers. That's its prime factorization. There's only one way to do it (aside from the order of the factors).

A factor tree is a handy way to find it: repeatedly split a number into two factors until every branch ends in a prime.

Example: Prime factorization of 60

Split 60 into any two factors — say, 6 and 10:

6 = 2 × 3 (both prime — done)
10 = 2 × 5 (both prime — done)

Collect all the primes at the ends of the branches:

$$60 = 2 \times 3 \times 2 \times 5 = 2^2 \times 3 \times 5$$

You'd get the same answer if you started differently (say, 4 × 15 or 3 × 20).

Example: Prime factorization of 84

84 = 2 × 42
42 = 2 × 21
21 = 3 × 7 (both prime)

$$84 = 2 \times 2 \times 3 \times 7 = 2^2 \times 3 \times 7$$

Example: Prime factorization of 100

100 = 10 × 10
10 = 2 × 5
10 = 2 × 5

$$100 = 2 \times 5 \times 2 \times 5 = 2^2 \times 5^2$$

Prime factorization is especially useful when finding the greatest common factor (GCF) or least common multiple (LCM) of two numbers.

Practice Problems

List all the factors of 24.

Show answerFactor pairs: $1 \times 24$, $2 \times 12$, $3 \times 8$, $4 \times 6$. All factors: 1, 2, 3, 4, 6, 8, 12, 24.

How many factors does 49 have?

Show answerFactor pairs: $1 \times 49$, $7 \times 7$. Factors: 1, 7, 49 — just three factors (49 is a perfect square).

Is 51 prime?

Show answerNo. $51 = 3 \times 17$. Both 3 and 17 are prime, making 51 composite.

Find the prime factorization of 72.

Show answer$72 = 8 \times 9 = 2^3 \times 3^2$. Check: $8 \times 9 = 72$ ✓

Find the prime factorization of 180.

Show answer$180 = 4 \times 45 = 4 \times 9 \times 5 = 2^2 \times 3^2 \times 5$. Check: $4 \times 9 \times 5 = 180$ ✓

You can also check your work with our Factoring Calculator.