Prime factorisation: every number in prime bricks — Mathematics, 11–13
Break a number into prime factors, by tree or by successive divisions, and see why there is only one way to do it.
Writing a number as a product of primes
Every natural number greater than 1 is either prime or can be written as a product of primes. For example, 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5. This is called prime factorisation: you keep breaking the number into factors until every factor is prime, and you gather the equal ones into a power. The order of the factors does not matter (5 × 3 × 2 × 2 is the same product), and each number has only one factorisation.
Why the factorisation is always the same
Primes are the bricks of the natural numbers: they cannot be broken any further. Break 60 in different ways: 6 × 10 = (2 × 3) × (2 × 5); 4 × 15 = (2 × 2) × (3 × 5); 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5. The paths differ, but every one ends with the same primes: two 2s, one 3 and one 5. That is why we say the factorisation is unique: it is like the number's identity card. No other number is made of 2² × 3 × 5.
Factorising 72 with a tree and with divisions
Tree: 72 = 8 × 9; then 8 = 2 × 4 = 2 × 2 × 2 and 9 = 3 × 3. Successive divisions, always by the smallest prime that works: 72 ÷ 2 = 36; 36 ÷ 2 = 18; 18 ÷ 2 = 9; 9 ÷ 3 = 3; 3 ÷ 3 = 1. The divisors used were 2, 2, 2, 3, 3. Both ways give 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3². Check: 8 × 9 = 72.
The trap: stopping too early
A reasonable slip: writing 72 = 8 × 9, or 72 = 2 × 2 × 2 × 9, and thinking you are done. Those are products, but 8 and 9 are not prime (8 = 2 × 2 × 2 and 9 = 3 × 3). Test each factor: a prime has only two divisors. Another slip is thinking that 2 × 2 × 2 × 3 × 3 and 3 × 2 × 3 × 2 × 2 are two different factorisations: only the order changed. And do not write 1 as a factor: it is not prime, and multiplying by 1 changes nothing.
Where you see it outside school: keeping data safe
Multiplying two primes is easy: 89 × 97 = 8633. But given only 8633, finding which two primes made it takes many tests of divisors. With primes hundreds of digits long, today's computers cannot do it in a useful time, and part of the protection of online messages and payments rests on that. And because the factorisation is unique, there is exactly one right answer to find.
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