Prime factorisation

We can break a number into factors until only primes are left. Take 56. It is composite: 56 = 4 × 14. Now 14 is composite too: 14 = 2 × 7, so 56 = 4 × 2 × 7. And 4 = 2 × 2, so 56 = 2 × 2 × 2 × 7. Now every factor is a prime, and we cannot break them further.

Writing a number as a product of primes like this is called its prime factorisation. The primes in it are its prime factors: the prime factors of 56 are 2 and 7. Every number greater than 1 has a prime factorisation. A prime like 7 is just 7. The number 1 has no prime factorisation at all.

An easy way to do it: divide by the smallest prime that works, again and again, until you reach 1. For 90: 90 ÷ 2 = 45, 45 ÷ 3 = 15, 15 ÷ 3 = 5, 5 ÷ 5 = 1. The primes you divided by give 90 = 2 × 3 × 3 × 5. Multiply them back and you get 90 again.

There are many ways to start, but they always end with the same primes. The book breaks 36 in four ways, starting from 2 × 18, 3 × 12, 4 × 9 and 6 × 6, and every time it gets two 2s and two 3s: 36 = 2 × 2 × 3 × 3. The order does not matter, because we can multiply in any order. We usually write the primes from smallest to largest, as in 225 = 3 × 3 × 5 × 5.

To factorise a product, factorise each part and put them together. 72 = 12 × 6, with 12 = 2 × 2 × 3 and 6 = 2 × 3. So 72 = 2 × 2 × 3 × 2 × 3, which is 2 × 2 × 2 × 3 × 3.

We can also go backwards. If a number's prime factorisation has one 2, two 3s and one 11, the number is 2 × 3 × 3 × 11 = 198.