# Question #99831

Dec 19, 2017

The LCM is $80$

#### Explanation:

Factor each number:

$16 = 2 \cdot 2 \cdot 2 \cdot 2 = {2}^{4}$

$20 = 2 \cdot 2 \cdot 5 = {2}^{2} \cdot 5$

Every multiple of $16$ has factors that include ${2}^{4}$

Every multiple of $20$ has factors that include ${2}^{2} \cdot 5$.

The Least Common Multiple must include factors of ${2}^{4}$, ${2}^{2}$ and $5$ and no extra factors (otherwise it wouldn't be "least")

The least number that includes these is ${2}^{4} \cdot 5 = 80$

Dec 19, 2017

$L C M = 2 \times 2 \times 2 \times 2 \times 5 = 80$

#### Explanation:

The LCM must be divisible by $16 \mathmr{and} 20$

Write each number as the product of its prime factors:

$\text{ } 16 = 2 \times 2 \times 2 \times 2$
$\text{ } \underline{20 = 2 \times 2 \textcolor{w h i t e}{\times \times \times} \times 5}$
$L C M = 2 \times 2 \times 2 \times 2 \times 5 = 80$

$80$ now contains all the factors of $16 \mathmr{and} 20$ but without any duplicates,