# Common multiple

A common multiple is a multiple shared by a given set of quantities, and a multiple is the product of a quantity and an integer. For example, the product of 2 and 6 is 12. The integer 12 can also be formed as the product of 3 and 4. Thus, 12 is a common multiple of both 2 and 3. A few other common multiples of 2 and 3 are 24, 108, and 1026.

Being able to find common multiples is important for working with fractions. In order to add or subtract fractions, the denominator of the fractions need to be the same. If they aren't the same, it is necessary to convert the fractions to equivalent fractions. This involves finding a common multiple for all the denominators, then multiplying each respective fraction by the appropriate constant in order to retain the value of the fraction.

Example

A common multiple of 2, 3, and 4, is 12. In order to solve the problem, the fractions above need to be converted to equivalent fractions with denominators of 12:

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In the above example, 12 is the least common multiple. When using the least common multiple to add or subtract a given set of fractions, the result will already be in simplest form. If, instead of using 12, the example above used 24, 36, 48, or some other multiple, the result would need to be simplified.

## How to find a common multiple

One of the simplest ways to find a common multiple between two (or more) numbers is to multiply all the numbers. The product of all the numbers will be a common multiple. However, it usually won't be the least common multiple.

Example

Find a common multiple of 2 and 8.

2 × 8 = 16

16 is a common multiple of 2 and 8, but not the least common multiple. 2 evenly divides 8, so 8 is a common multiple of both 2 and 8.

Another way to find a common multiple is to list multiples of each number until a common multiple is found. This method is straightforward but can be tedious depending on the numbers involved. The first common multiple found between the numbers (assuming all multiples up until the common multiple are listed) is the least common multiple.

Example

Find the first 3 common multiples of 3 and 4.

Multiples of 3:

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36...

Multiples of 4:

4, 8, 12, 16, 20, 24, 28, 32, 36...

12 is the least common multiple of 3 and 4. 24 and 36 are the next common multiples of 3 and 4. Notice that they are also multiples of 12. A multiple of a common multiple of two numbers is also a common multiple. There are therefore an infinite number of common multiples for a given set of numbers.

Another, more visual way to find a common multiple involves the use of a number line.

10 is the least common multiple of 2 and 5.