Numbers · Lesson 17 of 36

Highest Common Factor and Lowest Common Multiple

The HCF is the biggest number that divides both numbers. The LCM is the smallest number that both divide into. Listing or prime factors finds either.

Primary
Secondary
Updated 14 November 2026

What you will learn

Find the HCF and LCM of two numbers by listing.
Find them using prime factors.
Decide which one a problem needs.

Before you start: you should know factors, multiples and primes. See Factors and Multiples and Prime Numbers.

The idea

Two numbers can share factors and share multiples. The highest common factor (HCF) is the biggest number that is a factor of both. The lowest common multiple (LCM) is the smallest number that is a multiple of both.

Finding the HCF by listing

List the factors of each number, mark the ones they share, and pick the biggest.

Factors of 12 and 18. The shared factors are 1, 2, 3 and 6, and the biggest is 6, so HCF(12, 18) = 6.

Finding the LCM by listing

List multiples of each until they meet. The first number in both lists is the LCM.

Multiples of 4 are shaded and multiples of 6 are ringed. The first number with both is 12, so LCM(4, 6) = 12. The next is 24.

Using prime factors

For bigger numbers, write each as a product of primes.

36 split into primes: 2 × 2 × 3 × 3. For the HCF of 24 and 36, take the primes they share, using the smaller count of each: 2 × 2 × 3 = 12. For the LCM, use the larger count of each.

HCF: the primes in common, each used the fewest times. LCM: all the primes, each used the most times.

Which one do I need?

Sharing into equal groups, or simplifying a fraction: use the HCF.
Two things repeating together, or a common denominator: use the LCM.

Worked examples

Example 1: HCF with prime factors

Find the HCF of $24$ and $36$ .

$24 = 2 \times 2 \times 2 \times 3$ .
$36 = 2 \times 2 \times 3 \times 3$ .
Shared primes: $2, 2, 3$ .
$2 \times 2 \times 3 = 12$ .

Answer: $12$ .

Example 2: LCM with prime factors

Find the LCM of $12$ and $18$ .

$12 = 2^2 \times 3$ and $18 = 2 \times 3^2$ .
Take the most of each prime: $2^2$ and $3^2$ .
$4 \times 9 = 36$ .

Answer: $36$ .

Example 3: A timing problem

A light flashes every $9$ seconds and another every $12$ seconds. They flash together now. When do they flash together next?

This needs the LCM of $9$ and $12$ .
Multiples of $12$ : $12, 24, 36$ . Multiples of $9$ : $9, 18, 27, 36$ .
The first shared is $36$ .

Answer: after $36$ seconds.

The LCM is never smaller than the bigger number, and the HCF is never bigger than the smaller number.

Why it matters

The HCF simplifies fractions and splits things into the biggest equal groups, and the LCM finds common denominators and repeating schedules. Both appear again in Fractions and in algebra.

Practice

Your turn. Type or choose an answer, then press Check. Getting one wrong is fine, that is what practice is for. You can try as many times as you like.