Factor tree

Write the number at the top. Below it, draw two branches to any two numbers that multiply to make it, so 360 can split into 18 and 20. Do the same for each branch that isn't prime: 18 into 3 and 6, 20 into 4 and 5. Keep going until every branch ends in a prime, and circle those. The circled numbers multiplied together give the number at the top: 360 = 2 × 2 × 2 × 3 × 3 × 5.

Written with powers, that's 2³ × 3² × 5, sometimes called index form. The calculator splits each number into the two factors closest together, which is how most people would start and keeps the tree short; the answer at the bottom doesn't depend on that choice.

Start 360 as 10 × 36, or 2 × 180, and the tree looks different, but the circled primes are always three 2s, two 3s and one 5. That's the fundamental theorem of arithmetic: every whole number above 1 is a product of primes in exactly one way, apart from the order. It's why a factorisation can be checked by anyone, whichever tree they drew.

The primes also tell you how many factors a number has. Add 1 to each power and multiply, so 360 = 2³ × 3² × 5 has (3 + 1) × (2 + 1) × (1 + 1) = 24 factors, counting 1 and 360 itself.

Prime factors are the quickest way to the greatest common factor and least common multiple of a few numbers, and so to simplifying fractions and finding common denominators. The GCF and LCM calculators use them and draw the shared primes in a Venn diagram. They also tell you whether a number is a perfect square (every power even) or a cube (every power a multiple of 3).