Leave lesson

Number · Integers, powers & roots

1 / 8

Prime factorisation

Why every whole number breaks into a unique product of primes, how to find it with a factor tree, and how the prime form answers divisibility and square-number questions instantly.

Number · Integers, powers & roots

Prime factorisation

Why every whole number breaks into a unique product of primes, how to find it with a factor tree, and how the prime form answers divisibility and square-number questions instantly.

Why it works

Primes are the atoms of the whole numbers: a prime has exactly two factors (itself and 1), and every whole number above 1 is a product of primes in exactly one way. 360=23×32×5360 = 2^3 \times 3^2 \times 5 — no other bag of primes multiplies to 360. That uniqueness is why the prime form is so powerful: it's the number's fingerprint.

Why 1 is not a prime. If it were, factorisations wouldn't be unique — 360=23×32×5×1×1×…360 = 2^3 \times 3^2 \times 5 \times 1 \times 1 \times \ldots forever. One is the multiplicative wallpaper, not an atom; never list it as a factor.

The factor tree just keeps splitting. Break the number any way you like — 360=36×10360 = 36 \times 10 or 4×904 \times 90, it doesn't matter — and keep splitting anything that isn't prime until only primes hang from the tree:

360=36×10=(6×6)×(2×5)=2×3×2×3×2×5=23×32×5.360 = 36 \times 10 = (6 \times 6) \times (2 \times 5) = 2 \times 3 \times 2 \times 3 \times 2 \times 5 = 2^3 \times 3^2 \times 5.

Two rules close the job: every leaf must be prime (a 4 or a 15 left in the answer means keep splitting), and the final answer is a product in index form, not a list. Check by multiplying back.

The prime form answers questions at sight.
  • Divisibility: k=23×3×52k = 2^3 \times 3 \times 5^2 is divisible by 6 because
6=2×36 = 2 \times 3 and both atoms are present.
  • Squares: a square number's primes all carry even powers —
196=22×72=(2×7)2=142196 = 2^2 \times 7^2 = (2 \times 7)^2 = 14^2. To make k=23×3×52k = 2^3 \times 3 \times 5^2 into a square, top up the odd powers: multiply by 2×3=62 \times 3 = 6, giving 24×32×52=6022^4 \times 3^2 \times 5^2 = 60^2.
  • Scaling: if 2520=23×32×5×72520 = 2^3 \times 3^2 \times 5 \times 7, then
5040=2×2520=24×32×5×75040 = 2 \times 2520 = 2^4 \times 3^2 \times 5 \times 7 — one more 2, nothing else changes.