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

Take the prime factorization of $360$ . $360=2^3*3^2*5$ . We want $x$ to be as small as possible. And you want $x*360$ to be a square. So $x=2*5=10$ . $y$ is simmilar. $y=3*5^2=3*25=75$ So, $x+y=75+10=85$ , or $\boxed{\textbf{(B)}\ 85}$ .

~ModestFox97

Video Solution (XXX)

https://www.youtube.com/watch?v=ZuSJdf1zWYw ~David

2010 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 16	Followed by Problem 18
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2009 AMC 8 Problems/Problem 17

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Problem

Solution

Video Solution (XXX)

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