Prime numbers and factor trees

Years 5 to 9

Every number can be built from prime numbers, and a factor tree shows how.

A factor tree for 60. The green circles are the prime factors: 2, 2, 3 and 5.

For learners

A prime number can only be divided exactly by 1 and by itself. 2, 3, 5, 7, 11 and 13 are prime. 9 is not prime, because 3 × 3 = 9.

A factor tree breaks a number into smaller and smaller pairs until only primes are left. For 60: 60 = 6 × 10, then 6 = 2 × 3 and 10 = 2 × 5.

The primes at the ends of the branches are the prime factors. 60 = 2 × 2 × 3 × 5. Every number has just one set of prime factors.

A worked example

Write 72 as a product of prime factors.

72 = 8 × 9.
8 = 2 × 2 × 2 and 9 = 3 × 3.
So $72 = 2^{3} \times 3^{2}$ .

For grown-ups

Any whole number bigger than 1 is a product of primes in exactly one way (apart from the order). That is why prime factors are useful for finding the highest common factor and lowest common multiple.

How to help

Start by dividing by 2 while you can, then 3, then 5, then 7.
Write repeats as powers: $2 \times 2 \times 3 \times 5 = 2^{2} \times 3 \times 5$ .
Remember that 1 is not prime, and 2 is the only even prime.

Why it matters

Prime factors are used to simplify fractions, find common factors and multiples, and later in security codes that keep information safe online.

Topics in quarter that use this

Prime factors
Factors and multiples
Prime, square and cube numbers