LCM — Least Common Multiple

The least common multiple (LCM) of two numbers is the smallest number that both divide evenly.

It is the opposite kind of question to GCF. GCF goes down, LCM goes up.

Why it matters

LCM gives you the common denominator when you add or subtract fractions with different denominators. That's its whole job.

1/4 + 1/6   →   LCM(4, 6) = 12   →   3/12 + 2/12 = 5/12

Pick a bigger common denominator and the math still works, but you waste effort. The LCM is the smallest one that works.

How to find an LCM (the slow, sure way)

1. List the first several multiples of each number.
2. Look for the first number that appears in both lists.

Example 1: LCM(4, 6)

• Multiples of 4: 4, 8, 12, 16, 20, 24, 28, …
• Multiples of 6: 6, 12, 18, 24, …
• First common: 12
• LCM = 12

Example 2: LCM(5, 7)

• Multiples of 5: 5, 10, 15, 20, 25, 30, 35, …
• Multiples of 7: 7, 14, 21, 28, 35, …
• LCM = 35

When two numbers are coprime (like 5 and 7), the LCM is just their product.

Example 3: LCM(8, 12)

• Multiples of 8: 8, 16, 24, 32, …
• Multiples of 12: 12, 24, 36, …
• LCM = 24

Fast formula

For any two positive integers a and b:

LCM(a, b) = (a × b) / GCF(a, b)

So if you already know the GCF, you get the LCM almost for free.

Example: LCM(18, 24)

• GCF(18, 24) = 6
• LCM = (18 × 24) / 6 = 432 / 6 = 72

Sanity check: 72 ÷ 18 = 4 ✓, 72 ÷ 24 = 3 ✓.

Things to remember

• LCM(a, 1) = a always.
• LCM(a, a) = a always.
• LCM(a, b) ≥ max(a, b) always.
• If one number divides the other, the bigger one is the LCM.
  • LCM(6, 18) = 18 (because 6 divides 18).
• If a and b share no factors (coprime), LCM = a × b.

Try these (answers below)

1. LCM(3, 4)
2. LCM(6, 9)
3. LCM(10, 15)
4. LCM(12, 18)

1. 12 (3 × 4, coprime)
2. 18 (GCF=3, so LCM = 54/3)
3. 30
4. 36