*Posted by Arjun in : potd , trackback
*
Given that $a$, $b$, $x$, and $y$ are positive integers, and $a+b = 10$, find the number of ordered pairs $(a,b)$ such that: $$ a^{5\log_2x} = b^{\log_{32}y}$$ where the quotient of $\log x$ and $\log y$ is rational.

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Since $a+b$ is small, casework makes sense. However, one important point to notice is what happens when $a=b$:

$$ 5^{5\log_2x} = 5^{\log_{32}y} $$

$$ 5\log_2x = \log_{32}y $$

$$ \frac{5\log x}{\log 2} = \frac{\log y}{5 \log 2}$$

$$ \frac{\log x}{\log y} = \frac{1}{25}$$

By performing similar techniques when $a=2$ and $b=8$ and vice versa, it can be found that there are $\boxed{3}$ ordered pairs that satisfy this relation.

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