The first number both counts reach

A common multiple of two numbers is a multiple of each of them. Counting in 8s and in 12s, the first number both counts reach is the lowest common multiple, written LCM:

The lists stay short while the numbers are small. For 120 and 252 the multiples of 120 reach 2,520 at the 21st entry and the multiples of 252 at the 10th: 31 entries to reach the first number both lists hold.

Build a multiple of both from prime factors

Match the prime factors that 120 and 252 share. Keep every factor from 120, then add only the unmatched factors from 252; the result contains both prime factorizations and is the smallest common multiple.

Index form lets the primes be compared one at a time

In index form each index counts one prime: 2^³ in 120 is three factors of 2, and 2^² in 252 is two.

1. Factorise both numbers into index form, with prime bases in ascending order.
2. Compare the two indices of each prime, counting a missing prime as index 0, and take the higher.
3. Multiply the chosen prime powers. Every prime in either number is in the product.

Take every prime to its higher index

A multiple of 120 contains at least three 2s, a 3 and a 5. 120 × 7 = 840 = 2^³ × 3 × 5 × 7 still contains three factors of 2, and 2^² × 3 × 5 = 60 is not a multiple of 120, because it contains only two.

A common multiple of 120 and 252 contains every prime factor of each, so it holds each prime at least as many times as the number that holds it more: the higher index.

The smallest common multiple holds exactly the higher index of every prime, and one fewer is too few: 2^² × 3^² × 5 × 7 = 1,260 is not a multiple of 120, because 1,260 ÷ 120 = 10.5.

The LCM is never more than the product

120 × 252 = 30,240 is a multiple of 120 and of 252, so the LCM is at most 30,240. In prime factors, 30,240 = 2^⁵ × 3^³ × 5 × 7: every prime factor of 120 and every prime factor of 252.

Any shared prime makes the LCM smaller than the product.

Every common multiple is a multiple of the LCM

The common multiples of 120 and 252 are exactly the multiples of 2,520. A common multiple contains at least three 2s, two 3s, a 5 and a 7, so 2^³ × 3^² × 5 × 7 = 2,520 is a factor of it.

Find the LCM of two chosen numbers

Common mistakes

• Taking the lower index For 18 = 2 × 3^² and 20 = 2^² × 5, choosing the lower index of every prime gives 2, which is smaller than both numbers and a multiple of neither. The lower index belongs to the highest common factor, not the LCM.
• Keeping only the primes both numbers have For 18 and 20, taking 2 to its higher index and leaving out the 3s and the 5 gives 2^² = 4, a multiple of neither; the LCM is 2^² × 3^² × 5 = 180.
• Multiplying the two numbers 6 × 4 = 24 is a common multiple, and it is twice the lowest one, 12. The product is the LCM only when the two numbers share no prime.
• Writing the larger number 30 is the LCM of 12 and 30 only if 12 divides it, and 30 = 2 × 12 + 6; the LCM is 60.

Summary

1. A common multiple is a multiple of both numbers; the LCM is the smallest common multiple.
2. While the lists stay short, list the multiples of each number and pick the first that appears in both lists.
3. For larger numbers, write both in index form; the LCM is every prime in either number, each to its higher index: “find the lowest common multiple of 120 and 252” is answered 2,520.
4. The product of the two numbers is a common multiple, so the LCM is at most the product; the two are equal only when no prime is shared.
5. Every common multiple is a multiple of the LCM: the common multiples of 120 and 252 are the multiples of 2,520.
6. Divide the LCM by both numbers: both divisions are exact, and the quotients share no prime.