Lowest Common Multiple (LCM) · Hook

1/10

Ever wondered when two buses will leave the motor park together again?

At the motor park, a bus to Ibadan leaves every 4 minutes and a bus to Abeokuta leaves every 6 minutes. They both leave at 7:00 am. Emeka wants to know the next time they leave together — that is the lowest common multiple of 4 and 6.

Revision notes

📒 Lowest Common Multiple (LCM): lesson notes

Find the smallest number that two or more numbers all divide into — and use it to see when things happen together again.

At the motor park, a bus to Ibadan leaves every 4 minutes and a bus to Abeokuta leaves every 6 minutes. They both leave at 7:00 am. Emeka wants to know the next time they leave together — that is the lowest common multiple of 4 and 6.

Multiples and common multiples

The multiples of a number are its times table: the multiples of 4 are 4, 8, 12, 16, 20, 24, …

The multiples of 6 are 6, 12, 18, 24, 30, …

Numbers in both lists are common multiples: 12, 24, 36, … The smallest one, 12, is the LCM. So the buses leave together again after 12 minutes, at 7:12 am.

🔑 Key idea: The LCM is the smallest number that all the numbers divide into exactly.

A quick way: use the biggest number

List the multiples of the biggest number and stop at the first one the other numbers also divide.

For 8 and 12: 12 ✗ (8 does not divide 12), 24 ✓ (24 ÷ 8 = 3). So the LCM of 8 and 12 is 24.

Multiplying the numbers (8 × 12 = 96) gives a common multiple, but not always the lowest one.

🔑 Key idea: Check multiples of the biggest number in turn.

Using prime factors

Write each number in prime factors: 12 = 22 × 3 and 18 = 2 × 32.

Take every prime that appears, with its highest power: LCM = 22 × 32 = 4 × 9 = 36.

Don't mix it up with the HCF, which is a common factor and is never bigger than the numbers.

🔑 Key idea: LCM = each prime to its highest power, multiplied together.

Worked example

Find the LCM of 4 and 6.

  1. Step 1: Multiples of the biggest number, 6

    6, 12

  2. Step 2: Find the first common one

    12 ÷ 4 = 3

    12 ÷ 6 = 2

    LCM = 12

    Why? It is the smallest number every one of them divides into exactly.

Answer: 12

Summary

  • Multiples are the numbers in a times table.
  • The LCM is the smallest number that every one of the numbers divides into.
  • List multiples of the biggest number until the others divide it too.
  • With prime factors, take each prime to its highest power.
  • Use the LCM to find when repeating events next happen together.

Ready to check you've got it? Scroll up and work through the lesson: the interactive visuals, practice questions and mastery check are all there.