Ever wondered how to pack 12 mangoes and 18 oranges into identical bags?

Mama Nkechi wants every bag to have the same mix of fruit, with nothing left over, using as many bags as possible. The answer is hiding in the numbers 12 and 18 — it's their highest common factor.

Revision notes

📒 Highest Common Factor (HCF): lesson notes

Find the biggest number that divides two numbers exactly — and use it to share things into equal groups.

Mama Nkechi wants every bag to have the same mix of fruit, with nothing left over, using as many bags as possible. The answer is hiding in the numbers 12 and 18 — it's their highest common factor.

Factors and common factors

A factor of a number divides it exactly, with no remainder. The factors of 12 are 1, 2, 3, 4, 6 and 12.

The factors of 18 are 1, 2, 3, 6, 9 and 18.

Numbers that appear in both lists are common factors: 1, 2, 3 and 6.

🔑 Key idea: A common factor divides both numbers exactly.

The highest common factor

The highest common factor (HCF) is the biggest number in both lists. For 12 and 18 it is 6.

So Mama Nkechi can make 6 bags, each with 2 mangoes and 3 oranges.

For bigger numbers, write each as a product of primes and multiply the primes they share: 12 = 2 × 2 × 3 and 18 = 2 × 3 × 3 share 2 × 3 = 6.

🔑 Key idea: HCF = the product of the prime factors the numbers share.

Worked example

Find the HCF (highest common factor) of 10 and 45.

Step 1: Find a common factor
10 = 5 × 2
45 = 5 × 9
Why? 2 and 9 share no factor, so 5 is the biggest common factor.
Step 2: HCF
HCF = 5

Answer: 5

Summary

A factor divides a number exactly.
Common factors divide both numbers.
The HCF is the biggest common factor.
Use the HCF to split things into the largest number of equal groups.