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Number foundations

LCM: when numbers meet

Find the first shared multiple, understand why prime powers work, and solve repeating-event problems with confidence.

BeginnerAbout 12 min3 worked examples
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01 / THE STARTING POINT

Before you begin

What you need: Multiplication tables and prime factors.

  • Recognise a common multiple
  • Compare listing, prime factors and the HCF method
  • Model events that repeat at different intervals
02 / BUILD THE CONCEPT

Understand the idea

A shared destination

A multiple of a number is obtained by multiplying it by a positive whole number. The least common multiple is the smallest positive number in every list. For 4 and 6, that shared destination is 12.

Why the highest powers?

A common multiple must contain enough prime factors to be divisible by each input. Taking the highest power of every prime meets that requirement without adding unnecessary factors.

LCM versus HCF

HCF is the largest factor shared by the numbers; LCM is the smallest positive multiple shared by them. For positive integers a and b, their product equals HCF × LCM.

03 / A DIFFERENT WAY TO SEE IT

Two rhythms. One meeting point.

Multiples of 4
48121620
Multiples of 6
612182430
The highlighted 12 is the first positive stop shared by both sequences.
04 / FROM IDEA TO ANSWER

A method you can reuse

KEEP THIS HANDY

LCM(a, b) = a × b ÷ HCF(a, b)

For two positive integers. For three or more, combine two at a time.
  1. Write each number as a product of prime powers.
  2. Select the highest power of every prime appearing in any number.
  3. Multiply those powers and check that every original number divides the result.
A useful insight

If two positive integers are coprime, their HCF is 1, so their LCM is their product.

05 / WATCH THE METHOD WORK

Worked examples

Read the question first. Try a step yourself, then compare your reasoning.

EXAMPLE 01

Start with two small numbers

Find LCM(4, 6).

  1. Multiples of 4: 4, 8, 12, 16…
  2. Multiples of 6: 6, 12, 18…
  3. The first shared positive multiple is 12.
Answer12
EXAMPLE 02

Three repeating bells

Bells ring every 12, 18 and 30 minutes. They ring together at 9:00. When do they next coincide?

  1. 12 = 2² × 3; 18 = 2 × 3²; 30 = 2 × 3 × 5.
  2. LCM = 2² × 3² × 5 = 180 minutes.
  3. 180 minutes is 3 hours after 9:00.
Answer12:00 noon
EXAMPLE 03

One number divides another

Find LCM(8, 24).

  1. 24 is already a multiple of 8.
  2. Any common multiple must be at least 24.
  3. 24 is divisible by both inputs, so no larger candidate is needed.
Answer24
06 / YOUR TURN

Check your understanding

Quick questions, with explanations. These are for self-study and do not affect your account score.

0 of 3 questions checked

QUESTION 1Find LCM(8, 12).

See the explanation

24. 8 = 2³ and 12 = 2² × 3. Use 2³ × 3 = 24.

QUESTION 2Two lights flash every 6 and 10 seconds. After flashing together, when do they next coincide?

See the explanation

30 seconds. 6 = 2 × 3 and 10 = 2 × 5. LCM = 2 × 3 × 5 = 30 seconds.

QUESTION 3Find LCM(9, 27).

See the explanation

27. 27 is already divisible by 9. When one positive integer divides the other, the larger is the LCM.

07 / TAKE THE NEXT STEP

Put your understanding to work

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EXPLORE THE NUMBERS

LCM Calculator

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LCM Basics Quiz

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LCM Practice Quiz 1

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LCM Practice Quiz 2

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LCM practice

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