2009 AMC 8 Problems/Problem 17

Problem

The positive integers $x$ and $y$ are the two smallest positive integers for which the product of $360$ and $x$ is a square and the product of $360$ and $y$ is a cube. What is the sum of $x$ and $y$?

$\textbf{(A)}\   80    \qquad \textbf{(B)}\    85   \qquad \textbf{(C)}\    115   \qquad \textbf{(D)}\    165   \qquad \textbf{(E)}\    610$

Solution

The prime factorization of $360=2^3*3^2*5$. If a number is a perfect square, all of the exponents in its prime factorization must be even. Thus we need to multiply by a 2 and a 5, for a product of 10, which is x. Similarly, y can be found by making all the exponents divisible by 3, so $y=3*5^2=75$. Thus $x+y=\boxed{\textbf{(B)}\ 85}$.

See Also

2009 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 16
Followed by
Problem 18
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All AJHSME/AMC 8 Problems and Solutions

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