Primes and Prime Factorisation

A prime number has exactly two factors: 1 and itself. A composite number has more than two. Think of it as arranging dots into rectangles — seven dots only make a 1 by 7 line, so 7 is prime, while twelve dots make 3 by 4, 2 by 6 or 1 by 12, so 12 is composite. Two special cases catch people out. The number 1 is neither prime nor composite, because it has only one factor, not two. And 2 IS prime — it's the only even prime, because every other even number has 2 as an extra factor and so has at least three.

You don't have to test every possible divisor. Learn the quick checks: a number divides by 2 if its last digit is even; by 3 if its digits add to a multiple of 3; by 4 if the last two digits make a multiple of 4; by 5 if it ends in 0 or 5; by 6 if it passes both the 2 and the 3 test; by 9 if its digits add to a multiple of 9; and by 10 if it ends in 0. To decide whether a number is prime, you only need to try the primes up to its square root. For 97, test 2, 3, 5 and 7 — 11 times 11 is already 121, past 97, so there's nothing left to try. None divide it, so 97 is prime.

A factor tree breaks a number down into primes. Split 60 into 2 and 30, then 30 into 2 and 15, then 15 into 3 and 5. Circle each prime and stop splitting it. Reading the bottom of the tree gives 60 = 2 × 2 × 3 × 5, which is written 2² × 3 × 5 using indices. The powerful part is that it doesn't matter how you start — splitting 60 as 6 × 10 instead gives exactly the same primes at the bottom. Every whole number has one and only one prime factorisation, which is why this is such a reliable tool.

Prime factorisation gives you a reliable way to find the highest common factor and lowest common multiple, which beats listing factors once the numbers get big. Write both numbers as products of primes: 24 is 2³ × 3, and 36 is 2² × 3². For the HCF take the lowest power of each shared prime — 2² × 3, which is 12. For the LCM take the highest power of every prime that appears — 2³ × 3², which is 72. Lowest for HCF, highest for LCM. If you can't recall which way round it goes, reason it out: a common factor has to fit inside both numbers, so it can only take the smaller power.

This isn't a topic that sits on its own. Simplifying a fraction fully in one step means dividing top and bottom by their HCF. Adding fractions needs the lowest common denominator, which is just the LCM of the denominators. Simplifying surds works by spotting pairs in the prime factorisation — √72 is √(2³ × 3²), which gives 6√2. The index laws you met earlier operate on bases that are themselves built from primes. And worded problems like 'two bells ring every 12 and 18 minutes, when do they ring together?' are LCM questions in disguise. Primes are the building blocks, and every whole number is made of them in exactly one way.