How to find the lowest common multiple

Count up in twelves and in eighteens and you will eventually land on the same number. The first time that happens is the lowest common multiple. Britain says lowest, the United States says least, and both shorten it to LCM.

How to do it

1. Write the two numbers side by side, with a rule to the right of them.
2. Find a prime that divides at least one of them, and write it to the right of the rule.
3. Divide whichever numbers it goes into and carry the other one straight down, unchanged.
4. Keep going until both columns reach 1. Multiply every prime down the right-hand side — that is the LCM.

Worked examples

Using it to add fractions

The lowest common denominator is not a second thing to learn. It is the LCM of the two denominators, and finding it is the only hard part of adding fractions with different bottoms.

To add 1/4 and 1/6, find the LCM of 4 and 6, which is 12. Then rewrite both fractions over 12 — quarters become 3/12, sixths become 2/12 — and add the tops. The answer is 5/12.

Multiplying the denominators together also works and is easier to remember, but it makes the numbers bigger than they need to be. 1/4 and 1/6 over 24 gives 6/24 + 4/24 = 10/24, which then has to be simplified back to 5/12. The lowest common denominator saves that last step.

Two other ways, and when to use them

Listing is how the idea is introduced, and it is the only method that shows what the words mean. Multiples of 12 are 12, 24, 36, 48; multiples of 18 are 18, 36, 54. The first number in both lists is 36. It works for small numbers and stops being practical fast: the multiples of 63 and 84 do not meet until 252.

Prime factors give it without any listing. Break both numbers down, then take every prime that appears in either, as many times as it appears in whichever number has more of it. 12 = 2 × 2 × 3 and 18 = 2 × 3 × 3, so the LCM takes two 2s and two 3s — 36.

Why it works, and the check that catches errors

A common multiple has to contain everything both numbers are made of. Running the column to 1 and 1 empties both of them, so every prime either number needed is in the divisor list, and none is there more times than it had to be. That is what makes the answer the lowest common multiple rather than just a common one.

There is a neat check. Multiply the highest common factor by the lowest common multiple and you get the two original numbers multiplied together: for 12 and 18 that is 6 × 36 = 216, and 12 × 18 = 216. If the two answers disagree, one of them is wrong.

The other check is simpler and catches the commonest mistake outright: the LCM is never smaller than either number, and the HCF is never bigger than either. Getting them the wrong way round is the single most common error with this pair.

Where mistakes happen

• Multiplying the two numbers together and stopping. That is always a common multiple, but it is only the lowest one when the pair is coprime.
• Dividing the number the prime does not go into. It comes straight down unchanged, and changing it is what breaks the answer.
• Giving the highest common factor instead. If your answer is smaller than the numbers you started with, you have found the wrong one.

Questions people ask

Related

• How to find the highest common factor
• How to break a number into its prime factors
• How to tell if two numbers are coprime