# What is the least common multiple of 8 and 14?

Mar 7, 2018

See explanation...

#### Explanation:

Alright, first we need to look at the what an LCM is. A least common multiple (LCM) is the lowest multiple that two or more numbers share.

For example, the numbers $2$ and $8$, the LCM would be $8$ based on the following set of multiples:

Multiples of $2 :$ 2, 4, 6, 8, 10, 12, 14,...
Multiples of $8 :$ 8, 16, 24, 32, 40, 48, 56,...

So lets look at the LCM of $8$ and $14$ now. How I like to do it is you multiply the smaller number by $2$ until they are the same. They may seem confusing, let me explain.

Since $8$ is the smaller of the two numbers, once we multiply that by $2$ we get $16$. $16$ is not equal to $14$ so we keep going. To help you out, I've color coded the numbers so that $\textcolor{b l u e}{b l u e}$ is a multiple of $\textcolor{b l u e}{14}$ and $\textcolor{red}{red}$ is a multiple of $\textcolor{red}{8}$

$14 \cdot 2 = 28$
$\textcolor{b l u e}{28} \ne \textcolor{red}{16}$

$8 \cdot 3 = 24$
$\textcolor{red}{24} \ne \textcolor{b l u e}{28}$

$8 \cdot 4 = 32$
$\textcolor{red}{32} \ne \textcolor{b l u e}{28}$

$14 \cdot 3 = 42$
$\textcolor{b l u e}{42} \ne \textcolor{red}{32}$

$8 \cdot 5 = 40$
$\textcolor{red}{40} \ne \textcolor{b l u e}{42}$

$8 \cdot 6 = 48$
$\textcolor{red}{48} \ne \textcolor{b l u e}{42}$

$14 \cdot 4 = 56$
$\textcolor{b l u e}{56} \ne \textcolor{red}{48}$

$8 \cdot 7 = 56$
$\textcolor{red}{56} = \textcolor{b l u e}{56}$

Hope this helps!
~Chandler Dowd

Mar 7, 2018

56

#### Explanation:

To find the least common multiple find the prime factorization of each number and multiply the unique non repeated factors.

$8 = 2 \times 2 \times 2 = {2}^{3}$ There are 3 factors of 2 in 8

$14 = 2 \times 7 = {2}^{1} \times {7}^{1}$ There is 1 factor of 2 and 1 of 7 in 14

The 1 factor of 2 in 14 is a repeat of the 3 factors of 2 in 8. It is not
necessary to use that fourth factor of 2. three factors of 2 is enough

The 1 factor of 7 is a unique factor. The factor of 7 is necessary so

The LCM = $2 \times 2 \times 2 \times 7 = 56$

The least Common Multiple of 8 and 14 is 56