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Sum Of Squares And Multiple Of Product

Problem

Given that $x$, $y$ are positive integers and $x \ne y$, consider the following two results:

$5^2 + 13^2 = 194, 5 \times 13 \times 3 = 195$
$11^2 + 41^2 = 1802, 11 \times 41 \times 4 = 1804$

We can see that in both cases that the sum of the squares of $x$ and $y$ are almost a multiple of their product.

Prove that $x^2 + y^2$ can never be multiple of $xy$.

Problem ID: 300 (02 Jan 2007)     Difficulty: 3 Star

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