# 2013 AMC 8 Problems/Problem 10

## Problem

What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594? $\textbf{(A)}\ 110 \qquad \textbf{(B)}\ 165 \qquad \textbf{(C)}\ 330 \qquad \textbf{(D)}\ 625 \qquad \textbf{(E)}\ 660$

## Solution 1

To find either the LCM or the GCF of two numbers, always prime factorize first.

The prime factorization of $180 = 3^2 \times 5 \times 2^2$.

The prime factorization of $594 = 3^3 \times 11 \times 2$.

Then, to find the LCM, we have to find the greatest power of all the numbers there are; if one number is one but not the other, use it (this is $3^3, 5, 11, 2^2$). Multiply all of these to get 5940.

For the GCF of 180 and 594, use the least power of all of the numbers that are in both factorizations and multiply. $3^2 \times 2$ = 18.

Thus the answer = $\frac{5940}{18}$ = $\boxed{\textbf{(C)}\ 330}$.

## Similar Solution

We start off with a similar approach as the original solution. From the prime factorizations, the GCF is $18$.

It is a well known fact that $\gcd(m,n)\times \operatorname{lcm}(m,n)=|mn|$. So we have, $18\times \operatorname{lcm} (180,594)=594\times 180$.

Dividing by $18$ yields $\operatorname{lcm} (180,594)=594\times 10=5940$.

Therefore, $\dfrac{\operatorname{lcm} (180,594)}{\gcd(180,594)}=\dfrac{5940}{18}=\boxed{\textbf{(C)}\ 330}$.

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