Lowest common multiple

What is the lowest common multiple?

The LCM (lowest common multiple) of numbers is the smallest multiple that is common to both/all numbers.

In other words, it’s the smallest number that both/all numbers go into.

For example, the LCM of 12 and 16 is 48, because 48 is the smallest number that both 12 and 16 go into.

Finding the lowest common multiple

To find the lowest common multiple, you need to be familiar with prime factorisation.

Here are the steps:

1. Find the prime factorisation of each number.
2. For each different prime factor, take the highest power of that factor that appears in any of the numbers.
3. Multiply these together to get the LCM.

Example: Find the LCM of 18 and 24.

1. Find the prime factorisation of each number:
   • 18 = 2 \times 3 \times 3 = 2^1 \times 3^2
   • 24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3^1
2. For each different prime factor, take the highest power:
   • For 2, the highest power is 2^3 (from 24)
   • For 3, the highest power is 3^2 (from 18)
3. Multiply these together:
   • LCM(18, 24) = 2^3 \times 3^2 = 8 \times 9 = 72

Answer: 72

Example: Find the LCM of 36, 60 and 84.

1. Find the prime factorisation of each number:
   • 36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2
   • 60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3^1 \times 5^1
   • 84 = 2 \times 2 \times 3 \times 7 = 2^2 \times 3^1 \times 7^1
2. For each different prime factor, take the highest power:
   • For 2, the highest power is 2^2 (from all three numbers)
   • For 3, the highest power is 3^2 (from 36)
   • For 5, the highest power is 5^1 (from 60)
   • For 7, the highest power is 7^1 (from 84)
3. Multiply these together:
   • LCM(36, 60, 84) = 2^2 \times 3^2 \times 5^1 \times 7^1 = 4 \times 9 \times 5 \times 7 = 1260

Answer: 1260

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