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Greatest Common Factor

In one line: The greatest common factor of two numbers is the biggest whole number that divides into both of them exactly, with nothing left over.

Why it works

A factor is a whole number that divides into another number exactly. Finding the greatest common factor (GCF) helps you simplify fractions in one step, for example turning 24/36 into 2/3 by dividing the top and bottom by 12. It also solves real sharing problems, such as packing items into identical groups with nothing left over. Later, the same idea is used to factorise expressions in algebra.

Think of it like: Imagine you have 24 apples and 36 oranges, and you want to pack them into identical bags with nothing left over. Each bag must hold the same number of apples and the same number of oranges. The greatest common factor tells you the most bags you can make: 12 bags, each with 2 apples and 3 oranges. (Where this breaks: The bag picture only works for whole numbers you could actually count out, and it gets impractical with very large numbers. It also doesn't show the faster prime factor method, where you break numbers into primes (numbers with exactly two factors, 1 and themselves) and multiply the primes they share.)

Worked example

Find the greatest common factor of 24 and 36.

  1. List the factors of 24 in pairs: 1 × 24, 2 × 12, 3 × 8, 4 × 6. So the factors are 1, 2, 3, 4, 6, 8, 12, 24.
  2. List the factors of 36 in pairs: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6. So the factors are 1, 2, 3, 4, 6, 9, 12, 18, 36.
  3. Circle the factors that appear in both lists: 1, 2, 3, 4, 6 and 12.
  4. Pick the largest common factor: 12.
  5. Check with primes: 24 = 2 × 2 × 2 × 3 and 36 = 2 × 2 × 3 × 3. The shared primes are 2 × 2 × 3 = 12.

Answer: The greatest common factor of 24 and 36 is 12.

Common mistakes

  • Mixing up the GCF with the lowest common multiple and giving an answer bigger than the numbers, so remember the GCF can never be larger than the smaller number.
  • Stopping at the first common factor you spot (such as 6 for 24 and 36), so always list every factor and choose the largest one in both lists.
  • Missing factors when listing them, so work in pairs starting from 1 and stop when the pairs meet in the middle.
  • Multiplying all the prime factors of both numbers instead of only the shared ones, so take each prime only as many times as it appears in both numbers.

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What is the greatest common factor of 12 and 18?

Questions parents ask

What year do kids learn the greatest common factor?

Children usually meet factors and common factors in Years 5–6 in the UK and Australia, then find the highest common factor more formally in Years 6–8. In the US, the greatest common factor is a standard Grade 6 topic. It builds on secure times tables knowledge.

Is the greatest common factor the same as the highest common factor?

Yes, they are exactly the same thing with different names. The UK and Australia usually say highest common factor (HCF), while the US usually says greatest common factor (GCF). You may also see it called the greatest common divisor (GCD).

How can I help my child with the greatest common factor at home?

Practise times tables first, because finding factors depends on them. Then try sharing problems with real objects, like splitting sweets and biscuits into identical bags, and ask how many bags you can make. Encourage your child to list factors in pairs so none are missed.

What is the difference between the greatest common factor and the lowest common multiple?

The greatest common factor is the biggest number that divides into both numbers, so it is never larger than the smaller number. The lowest common multiple is the smallest number that both numbers divide into, so it is never smaller than the larger number. For 4 and 6, the GCF is 2 and the lowest common multiple is 12.