Prime Factors, HCF & LCM

• Prime factor decomposition (PFD) — breaking a number down into the prime numbers that multiply to make it, usually with a factor tree.
• Index notation — writing repeated prime factors as powers, e.g. 360 = 2³ × 3² × 5.
• Spotting squares & cubes — if every power in the PFD is even, it's a square number; if every power is a multiple of 3, it's a cube number.
• Square roots via PFD — halving the powers gives you the square root instantly (and works for exact/surd answers too).
• HCF (Highest Common Factor) — the biggest number that divides two (or more) numbers exactly. Found using the lowest shared powers of common primes.
• LCM (Lowest Common Multiple) — the smallest number that both numbers divide into exactly. Found using the highest power of every prime that appears anywhere.
• Venn diagrams — a visual way to sort common vs. non-common prime factors, making both HCF and LCM easy to read off.

What actually is a prime factor?

Let's build this from the ground up. A factor of a number is anything that divides into it with nothing left over. So the factors of 18 are 1, 2, 3, 6, 9, and 18 — because each of those divides 18 exactly.

A prime number is a special, stubborn kind of number: it has exactly two factors — 1 and itself. Nothing else divides into it. The first primes are:

Notice 2 is the only even prime — every other even number can be divided by 2, so it fails the "exactly two factors" test. And 1 is not prime, because it only has one factor (itself) — it fails the definition too.

Now here's the powerful idea: if you take any whole number bigger than 1 and keep breaking it into factors, breaking those into factors, and so on — you will always end up at the same set of prime numbers, no matter which path you took to get there. This is sometimes called the "Fundamental Theorem of Arithmetic" — but forget the fancy name, just remember: every number has one unique prime "recipe."

Method: The Factor Tree

The cleanest way to find prime factors is a factor tree. You split your number into any two factors, then keep splitting each branch until every "leaf" is a prime number (circle these — they're your answer).

Example: Find the prime factors of 360

Writing it with index notation

Once you have all the primes listed out, tidy the answer up using powers (indices) for any prime that repeats:

ascending order