• 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

Reading off the circled primes: 360 = 2 × 2 × 2 × 3 × 3 × 5. It doesn't matter whether you started by splitting 360 into 2 × 180 or into 10 × 36 — you always land on the exact same set of circled primes at the end. Try it yourself with 10 × 36 and you'll get there too.

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:

Always write the primes in ascending order (smallest first) — examiners expect this and it makes your working easy to check.

Spotting square and cube numbers instantly

Here's a beautiful pattern once you have a number's PFD in index form:

Why does this work? Think about squaring something like (2² × 3): when you square it, every power inside doubles → (2² × 3)² = 2⁴ × 3². Every exponent becomes even. So working backwards, if you see a number whose PFD has all even powers, you know it must be "something squared."

Example: 7056 = 2⁴ × 3² × 7². All three powers (4, 2, 2) are even, so 7056 is a square number — specifically, 7056 = (2² × 3 × 7)² = 84².

Example: 1 728 000 = 2⁹ × 3³ × 5³. All three powers (9, 3, 3) are multiples of 3, so it's a cube number: 1 728 000 = (2³ × 3 × 5)³ = 120³.

Finding square roots using PFD

If a number is a perfect square, you can find its exact square root without a calculator by simply halving every power in its PFD.

Worked Example: Find √144.

144 = 2⁴ × 3² → halve the powers → 2² × 3 → multiply out → 4 × 3 = 12. ✓ (Check: 12² = 144)

The trickier bit: exact square roots of non-square numbers

Sometimes a number isn't a perfect square (its powers aren't all even), but you can still simplify its square root into "surd" form — a whole number times √(something).

Worked Example: Find the exact square root of 1440.

What is the HCF?

A common factor of two numbers is any value that divides exactly into both of them. 1 is always a common factor of any two numbers (it's the "lowest common denominator" of factors, in a sense). The Highest Common Factor is simply the biggest of all these shared factors — useful for simplifying fractions and factorising expressions.

Method A: Listing factors (fine for small numbers)

List every factor of each number, then pick the largest one that appears in both lists. Simple, but slow and error-prone for big numbers — which is why PFD is the better tool.

Method B: Using prime factor decomposition (the smart way)

Write both numbers in PFD form. For every prime that appears in both lists, take the lower of the two powers. Multiply these together — that's your HCF.

Worked Example: Find the HCF of 24 and 60.

24 = 2³ × 3    60 = 2² × 3 × 5

Common primes: 2 (lowest power is 2² since 24 only has 2³, 60 has 2²) and 3 (lowest power is 3¹, matching in both). 5 doesn't appear in 24, so it's excluded entirely.

Method C: The Venn diagram approach

This is brilliant for visual thinkers. Draw two overlapping circles, one per number. Write each number's prime factors as individual entries (not powers) — if a prime is shared, it goes in the overlap; anything left over goes in the outer part of its own circle.

Example: 42 = 2 × 3 × 7 and 90 = 2 × 3 × 3 × 5

The HCF is the product of everything in the overlap: HCF = 2 × 3 = 6. The LCM is the product of every number in the whole diagram (all three circles' contents): LCM = 7 × 2 × 3 × 3 × 5 = 630. One diagram gives you both answers!

What is the LCM?

A common multiple of two numbers is a number that appears in both of their times tables. The product of the two original numbers is always a common multiple — but usually not the smallest one. The Lowest Common Multiple is the smallest number that both numbers divide into exactly — this is exactly what you use when adding or subtracting fractions with different denominators.

Method: Using prime factor decomposition

Write both numbers in PFD form. This time, for every prime that appears in either number (they don't need to be shared!), take the highest power it appears with anywhere. Multiply these together.

Worked Example: Find the LCM of 72 and 540.

72 = 2³ × 3²    540 = 2² × 3³ × 5

Highest power of 2 seen anywhere: 2³ (from 72). Highest power of 3 seen anywhere: 3³ (from 540). Highest power of 5 seen anywhere: 5¹ (from 540, even though 72 has no 5 at all — it still counts!).

• I can define a factor and a prime number correctly.
• I can build a factor tree to find the prime factors of any number.
• I can write a prime factor decomposition using index notation, primes in ascending order.
• I can identify whether a number is a perfect square or cube just by looking at its PFD indices.
• I can find the exact square root of a perfect square using PFD (halving the powers).
• I can find the exact square root (surd form) of a non-square number using PFD.
• I can find the HCF of two numbers using PFD (common primes, lowest power).
• I can find the LCM of two numbers using PFD (all primes, highest power).
• I can draw and correctly read a Venn diagram to find both HCF and LCM.
• I know I do NOT mix up the HCF and LCM rules (lowest vs highest power, common vs all primes).