Guides for parents · 1 November 2026

What are prime factors? A guide for parents

Prime factors are the prime numbers that multiply together to make a number. 60 is 2 × 2 × 3 × 5, and no other set of primes makes 60. That one fact sits underneath simplifying fractions, HCF and LCM, and a good part of secondary-school algebra.

Three words first

Factors of a number divide into it exactly. The factors of 12 are 1, 2, 3, 4, 6 and 12. They come in pairs that multiply to 12: 1 × 12, 2 × 6 and 3 × 4.

A prime number has exactly two factors: 1 and itself. The primes below 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. A number with more than two factors, such as 12, is called composite.

1 is not prime, because it has only one factor. This is not a technicality. If 1 counted as prime, 12 could be written as 2 × 2 × 3, or 1 × 2 × 2 × 3, or 1 × 1 × 2 × 2 × 3, and the "only one way" fact below would stop being true.

So the prime factors of 12 are the factors that are prime: 2 and 3. Writing 12 as a product of primes gives 12 = 2 × 2 × 3.

Every number has exactly one prime recipe

Every whole number above 1 can be written as a product of primes in exactly one way, apart from the order. Mathematicians call this the fundamental theorem of arithmetic. Children can think of primes as the building blocks, and of each number as one fixed recipe.

That is why two children who split 60 differently still end up in the same place.

Finding them with a factor tree

Split the number into any factor pair, then keep splitting until every branch ends in a prime.

      60                      60
     /  \                    /  \
    6    10                 4    15
   / \   / \               / \   / \
  2   3 2   5             2   2 3   5

60 = 2 × 2 × 3 × 5        60 = 2 × 2 × 3 × 5

Ring each prime as you reach it so nothing gets lost. Write the answer in order, smallest first: it makes checking and comparing much easier.

Or divide by the smallest prime

Some children prefer a ladder: keep dividing by the smallest prime that goes in exactly.

84 ÷ 2 = 42
42 ÷ 2 = 21
21 ÷ 3 = 7      7 is prime: stop

84 = 2 × 2 × 3 × 7

The divisibility rules make the "does 3 go in?" step quick.

Index notation

Repeated primes are written with powers: 60 = 2² × 3 × 5, and 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5. In England, children meet the words prime, prime factor and composite in Year 5, and usually write full prime factorisations with powers at secondary school.

The three mistakes to watch for

Stopping too early. 12 = 2 × 6 is a factor pair, not a prime factorisation: the 6 still splits into 2 × 3.
Losing a repeat. "The prime factors of 12 are 2 and 3" is right, but "12 = 2 × 3" is not. Multiply the answer back out to check.
Counting 1, or 9, as prime. 1 has only one factor, and 9 = 3 × 3. Odd does not mean prime.

What it is used for

Simplifying fractions: 24/36 = (2 × 2 × 2 × 3) / (2 × 2 × 3 × 3). Cross out the shared 2 × 2 × 3 and 2/3 is left.
HCF and LCM come straight from the two factor trees.
Counting factors: 60 = 2² × 3 × 5 has (2 + 1) × (1 + 1) × (1 + 1) = 12 factors, without listing them.

Try it at home

Pick a number on a bus or a door. Each of you grows a factor tree your own way, then compare the primes at the bottom.
Try to arrange 7 or 11 counters into a rectangle with more than one row. When you can't, the number is prime.