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

This is very easy. To find the LCM of 180 and 594, first find the prime factorization of both.

The prime factorization of 180 = $3^2 \times 5 \times 2^2$ The prime factorization of 594 = $3^3 \times 11 \times 2$

Then, 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 and multiply. $3^2 \times 2$ = 18.

Thus the answer = $\frac{5940}$ (Error compiling LaTeX. Unknown error_msg) = $\boxed{\textbf{(C)}\ 330}$

2013 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
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2013 AMC 8 Problems/Problem 10

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