# 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?

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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 $12 \times 45 = 540$

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sjc Share
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

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

$84 b = 12 \times 3780$

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

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

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Feb 12, 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

$45 \times 12 = 540$

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