# What is the least common multiple (LCM) of 8, 12, and 18?

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Jul 6, 2016

Least Common Multiple is $72$

#### Explanation:

Multiples of $8$ are $\left\{8 , 16 , 24 , 32 , 40 , 48 , 56 , 64 , 72 , 80 , 88 , 96 , 104 , 112 , 120 , 128 , 136 , 144. \ldots . .\right\}$

Multiples of $12$ are $\left\{12 , 24 , 36 , 48 , 60 , 72 , 84 , 96 , 108 , 120 , 132 , 144. \ldots . .\right\}$

Multiples of $18$ are $\left\{18 , 36 , 54 , 72 , 90 , 108 , 126 , 144. \ldots . .\right\}$

Hence common multiples are $\left\{72 , 144 , \ldots \ldots .\right\}$

and Least Common Multiple is $72$

Another shorter way is to write numbers as multiplication of its prime factors

$8 = 2 \times 2 \times 2$ $\to$ $2$ comes three times

$12 = 2 \times 2 \times 3$ $\to$ $2$ comes twice and $3$ once.

$18 = 2 \times 3 \times 3$ $\to$ $2$ comes once and $3$ twice

So maximum times is three times for $2$ and two times for $3$.

Hence, Least Common Multiple is $2 \times 2 \times 2 \times 3 \times 3 = 72$.

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