Common Multiple — Definition, Formula & Examples

A common multiple is a whole number that appears in the multiplication tables of two or more numbers. For example, 12 is a common multiple of 3 and 4 because 3 × 4 = 12 and 4 × 3 = 12.

An integer $m$ is a common multiple of integers $a$ and $b$ if $a \mid m$ and $b \mid m$ —that is, if $m$ is divisible by both $a$ and $b$ with no remainder. Every pair of nonzero integers has infinitely many common multiples.

How It Works

To find common multiples of two numbers, list several multiples of each number and look for values that appear in both lists. For instance, multiples of 4 are 4, 8, 12, 16, 20, 24, … and multiples of 6 are 6, 12, 18, 24, 30, … so the common multiples include 12, 24, 36, and so on. The smallest of these is called the least common multiple (LCM). Every other common multiple is a whole-number multiple of the LCM.

Worked Example

Problem: Find the first three common multiples of 6 and 8.

Step 1: List multiples of 6:

6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, \ldots

Step 2: List multiples of 8:

8, 16, 24, 32, 40, 48, 56, 64, 72, \ldots

Step 3: Identify numbers that appear in both lists:

24,\; 48,\; 72,\; \ldots

Answer: The first three common multiples of 6 and 8 are 24, 48, and 72.

Why It Matters

Finding common multiples is essential when you add or subtract fractions with different denominators—you need a common denominator, which is a common multiple. This skill also comes up in scheduling and pattern problems, such as figuring out when two events with different cycles will coincide.

Common Mistakes

Mistake: Confusing common multiples with common factors. Students sometimes look for shared divisors instead of shared multiples.

Correction: Common factors divide into both numbers (and are smaller or equal), while common multiples are divisible by both numbers (and are larger or equal). For 6 and 8, the greatest common factor is 2, but the least common multiple is 24.