# Least Common Multiples (LCMs)

A common multiple of two whole numbers $a$ and $b$ is a number $c$ which $a$ and $b$ both divide into evenly.

For example, $48$ is a common multiple of $6$ and $12$ since

$48÷6=8$   and

$48÷12=4$.

The least common multiple is just what it sounds like... the smallest of all the common multiples.

Example 1:

Find the least common multiple of $9$ and $12$.

To do this, we can list the multiples:

$\begin{array}{l}9:9,18,27,\underset{_}{36},45,54,63,72,...\\ 12:12,24,\underset{_}{36},48,60,72,...\end{array}$

$36$ is the first number that occurs in both lists. So $36$ is the LCM.

The listing method is impractical for large numbers. Another way to find the LCM of two numbers is to divide their product by their greatest common factor (GCF).

Example 2:

Find the least common multiple of $18$ and $20$.

To find the GCF of $18$ and $30$, you can write their prime factorizations:

$\begin{array}{l}18=2\cdot 3\cdot 3\\ 30=2\cdot 3\cdot 5\end{array}$

The common factors are $2$ and $3$. So, the GCF is $2\cdot 3=6$.

Now find the LCM by multiplying the two numbers and dividing by the GCF. (You can make this calculation a little easier by cancelling a common factor.)

$\frac{18\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}30}{6}=\frac{3\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}6\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}30}{6}=90$

When the GCF of two numbers is $1$, the LCM is equal to the product of the two numbers.

Example 3:

Find the least common multiple of $10$ and $27$.

$10$ and $27$ share no common factors other than $1$. So, the GCF is $1$.

Therefore, the LCM is simply $10\cdot 27=270$.

A third way to find the LCM of is to list all of the prime factors of each number and then multiply all of the factors the greatest number of times each occurs in any of the lists. [Note that while the previous method won't always work with more than $2$ numbers, this method will.]

Example 4:

Find the LCM of $16,25$ and $60$.

$\begin{array}{l}16=2\cdot 2\cdot 2\cdot 2\\ 25=5\cdot 5\\ 60=2\cdot 2\cdot 3\cdot 5\end{array}$

The greatest number of times the factor $2$ occurs is four (in the first list).

The greatest number of times the factor $3$ occurs is one (in the third list).

The greatest number of times the factor $5$ occurs is two (in the second list).

So, we multiply four $2$s, one $3$, and two $5$s.

LCM $=2\cdot 2\cdot 2\cdot 2\cdot 3\cdot 5\cdot 5=1200$