Prime factorisation — Mathematics, 11–13 years
Prime factorisation breaks a whole number into the prime numbers that multiply to make it. It gives each number a useful fingerprint for comparing, simplifying and organising numbers.
The building blocks of a number
A prime number has exactly two factors: 1 and itself. Prime factorisation writes another whole number as a multiplication of primes, such as 60 = 2 × 2 × 3 × 5. The order does not matter, because multiplication can be rearranged.
Why break numbers apart?
Large numbers can hide how they are related. Splitting them into primes reveals shared building blocks, so we can spot common factors and simplify calculations without guessing. Mathematicians developed this way of looking at numbers because multiplication is often easier to compare than large totals.
Factoring 84
Start with 84, which is even, so divide by 2: 84 = 2 × 42. Divide 42 by 2 again: 42 = 2 × 21. Then 21 = 3 × 7, and both 3 and 7 are prime. Put every step together: 84 = 2 × 2 × 3 × 7.
Stopping too soon
A common mistake is to stop after finding factors such as 84 = 4 × 21. That answer is correct as a factorisation, so the mistake is understandable, but 4 and 21 are not prime. Keep splitting every factor until each one has only 1 and itself as divisors.
Where it is useful
Prime factors help software create secure codes, because multiplying primes is quick but reversing the process for a huge number is difficult. They also help when sharing objects into equal groups or reducing a fraction. In both cases, the hidden building blocks tell us what divisions are possible.
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