LCM by Common Division Method

How to find the LCM by common division

The LCM (Lowest Common Multiple) of two or more numbers is the smallest number that all of them divide into. The common division method finds it quickly: keep dividing by the smallest prime that goes into at least one of the numbers, until every number becomes 1. Use the animation above to watch each step for 20 problems, then follow the written method.

The method — step by step

1. Write the numbers in a row.
2. Divide by the smallest prime (2, then 3, 5, 7 …) that divides at least one of them. Divide the numbers it goes into; carry the others down unchanged.
3. Repeat with the new row, always choosing the smallest prime that still works.
4. Stop when every number is 1.
5. The LCM is the product of all the primes in the left column.

Worked example: LCM of 12 and 18

Divide by	Numbers
2	12, 18
2	6, 9  (9 is carried down)
3	3, 9
3	1, 3  (1 is carried down)
—	1, 1

LCM = 2 × 2 × 3 × 3 = 36.

LCM vs HCF — the difference

In this method for LCM, you divide when the prime goes into at least one number. For the HCF (Highest Common Factor) you would only divide when the prime goes into every number, and the HCF is the product of just those common primes.

Common mistakes to avoid

• Stopping too early — keep going until every number is 1.
• Dropping a number that isn't divisible — carry it down unchanged instead.
• Using the common primes only (that gives HCF, not LCM).

Practice problems

Try these in the animation above: 8 and 12, 10 and 15, 16 and 24, 9 and 15, 20 and 30, 14 and 35.

Quick tip

The LCM is useful whenever you add fractions — it's exactly the common denominator you need. Finding the LCM well makes fraction work much easier.