# What is the least common multiple of 8, 10, 12?

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79
Dec 6, 2017

Least Common Multiple is $120$

#### Explanation:

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

Multiples of $10$ are $\left\{10 , 20 , 30 , 40 , 50 , 60 , 70 , 80 , 90 , 100 , 110 , \textcolor{red}{120} , \ldots .\right\}$

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

Hence common multiples are $\left\{120 , 240 , - - - -\right\}$

and Least Common Multiple is $120$

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2
Jan 10, 2018

The general rule is to decompose the numbers into prime factors and then the lcm in the product of all factors raised to the highest power found in those numbers.

$8 = {2}^{3}$

$10 = 2 \times 5$

$12 = {2}^{2} \times 3$

$\lcm \left(8 , 10 , 12\right) = {2}^{3} \times 3 \times 5 = 120$

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