# There are two numbers. One of them is 84. Their Lowest Common Multiple (LCM) is 3780 and their Greatest Common Factor (GCF) is 12. What is the other number?

Feb 11, 2018

The second number would be the product of 45 and 12
540

#### Explanation:

Dividing 3780 by 84 gives one of the common factors of 84 and the second number.

$\frac{3780}{84} = 45$

The HCF = 12 so the second number has factors of 45 and 12

$45 \times 12 = 540$

Feb 12, 2018

$540$

#### Explanation:

we have

$h c f \left(84 , b\right) = 12$

$\lcm \left(84 , b\right) = 3780$

to find $b$

we have the well known relationship

$a b = h c f \left(a , b\right) \times \lcm \left(a , b\right)$

$84 b = 12 \times 3780$

$b = \frac{\cancel{12} \times 3780}{\cancel{84}} ^ 7$

$b = \frac{3780}{7} = 540$

Feb 12, 2018

The other number is $540$

#### Explanation:

For any questions involving HCF and LCM,find each number as the product of the prime factors. That will tell you what you are working with.

$\text{ } 84 = \textcolor{b l u e}{2 \times 2 \times 3} \times \textcolor{red}{7}$
" "? = ul(color(blue)(?xx?xx?)xx?xx?xx?xx?)

$H C F = \textcolor{b l u e}{2 \times 2 \times 3} \textcolor{w h i t e}{w w w w w w w w w w w w w w} = 12$
$L C M = \textcolor{b l u e}{2 \times 2 \times 3} \times \textcolor{red}{7} \times \textcolor{p u r p \le}{3 \times 3 \times 5} \text{ } = 3780$

$3780 = \textcolor{b l u e}{H C F} \times \textcolor{red}{7} \times \text{ another number}$
$3780 = \text{ } \textcolor{b l u e}{12} \times \textcolor{red}{7} \times \textcolor{p u r p \le}{\left(3 \times 3 \times 5\right)}$
$3780 = \text{ } \textcolor{b l u e}{12} \times \textcolor{red}{7} \times \textcolor{p u r p \le}{45}$

The other number cannot have $7$ as a factor otherwise the $H C F$ would be $84$

The other number is therefore $12 \times 45 = 540$