What is the least common multiple?

Before discussing least common multiples, let us discuss multiples and common multiples.

A multiple is a number that can be obtained by multiplying an integer by a whole number.

For example, a multiple of 2 is 2,4,6,8,10,12,..., and a multiple of 3 is 3,6,9,12,15,16,....

A common multiple is a multiple that is common to two or more integers.

For example, multiples of 3 are 3,6,9,12,15,..., and multiples of 5 are 5,10,15,20,25,30,..., so multiples of 3 and 5 are 15,30,...

The least common multiple is the smallest number among the common multiples.

Since the common multiple of 3 and 5 is 15,30,..., the smallest number, 15, is the least common multiple.

We can see that the common multiple is an integer multiple of the least common multiple.

How to find the lreatest common multiple

Least common multiples can be computed using prime factorization.

Here we will compute the least common multiple of 12 and 90 as an example.

First, we prime factorize these two numbers.

\begin{align} 90&=2\times3^{2}\times5\notag\\ 12&=2^{2}\times3\notag \end{align}

Next, we compare how many of each prime factor is contained in each number and take out the larger number.

For ease of comparison, the prime factorized expressions are expressed as follows.

\begin{align} 90&=2^{1}\times3^{2}\times5^{1}\notag\\ 12&=2^{2}\times3^{1}\times5^{0}\notag \end{align}

90 contains one 2 and two 12s, so we take out \(\displaystyle 2^{2} \).

90 contains two 3s and one 12, so we take out \(\displaystyle 3^{2} \).

90 contains one 5 and zero 12, so we take out \(\displaystyle 5^{1} \).

Finally, multiply all the extracted numbers together to obtain the least common multiple.

\[ 2^{2}\times3^{2}\times5^{1}=180 \]

Example

Example 1

Find the least common multiple of 84 and 90.

Solution

Prime factorize 84 and 90 as follows.

\begin{align} 84&=2^{2}\times3\times7\notag\\ 90&=2\times3^{2}\times5\notag \end{align}

Compare the exponents of each prime factor and multiply by the larger number to obtain.

\[ 2^{2}\times3^{2}\times5\times7=1260 \]

Example 2

Find the least common multiple of 75 and 95.

Solution

Prime factorize 75 and 95 as follows.

\begin{align} 75&=3\times5^{2}\notag\\ 95&=5\times19\notag \end{align}

Compare the exponents of each prime factor and multiply by the larger number to obtain.

\[ 3\times5^{2}\times19=1425 \]