Prime Factorisation

Every composite number can be taken apart into prime pieces. Start with twelve. Split it into three times four. Three is prime, so it is finished, but four is not, so split that again into two times two. Now every piece is prime and there is nothing left to break, which means twelve is two times two times three. Try it on any composite you like and the same thing happens: six is two times three, eight is two times two times two, twenty is two times two times five. A prime is already as broken down as it can get, so there is nothing to do to it. This taking-apart is called prime factorisation, and it is why primes are called the building blocks of number.

A factor tree is just a tidy way of writing that splitting down. Put thirty-six at the top. Choose any factor pair you like — say four times nine — and draw two branches. Neither four nor nine is prime, so each one branches again: four into two and two, nine into three and three. The moment a number turns prime, circle it and stop branching there. When every branch ends in a prime the tree is complete, and you read the answer straight off the bottom row: thirty-six equals two times two times three times three. Always check by multiplying that bottom row back together. Two times two is four, four times three is twelve, and twelve times three is thirty-six.

Here is the surprising part. It does not matter which factor pair you pick at the start. Break seventy-two as eight times nine: the eight gives two, two and two, and the nine gives three and three. Break it instead as six times twelve: the six gives two and three, and the twelve gives two, two and three. Different trees, different branches, but line the leaves up in order and both come out as three twos and two threes. Every whole number has exactly one prime factorisation, however you get there. That is a genuinely deep fact about numbers, and for you it means something very practical: start with whichever factor pair you happen to spot first, and you will still be right.

Writing two times two times two times three times three is fine, but it gets long. Indices tidy it up. Count how many of each prime you have — three twos and two threes — and write two to the power three, times three to the power two. The small raised number tells you how many of that prime there are, and nothing else. Look at the examples: eight is three twos, so 2³. Forty is three twos and a five, so 2³ × 5. One hundred is two twos and two fives, so 2² × 5². A prime that turns up only once needs no index at all — you would never write 5¹. And always list your primes smallest first.

Two habits will keep you right. First, always multiply back. If you claim ninety is two times three squared times five, work it out: three squared is nine, two nines are eighteen, and eighteen fives are ninety. It checks out. Second, if trees get messy, use the division ladder instead. Write ninety down, divide by the smallest prime that goes into it, and put the answer underneath. Ninety divided by two is forty-five. Forty-five divided by three is fifteen. Fifteen divided by three is five. Five divided by five is one. Stop at one, and the primes running down the left-hand side are your answer: two times three squared times five.