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$B=\left\{\left(x,y\right)\mid x^{2}+y^{2}=625 \quad x,y\in \mathbb N\right\}$
What is the number of elements of $B$?

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1 Answer

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$x^2 + y^2 = 625 \implies x^2 = 625 -y^2 \implies x^2 = (25-y)(25+y)$
Hence $(25-y)(25+y)$ should be square number and then by checking $y=1,2,...,25$ we can deduce :
$y = 7 , 15 , 20 , 24 $
Because other possibilities are remove quickly.
For example if $y=11$ then $25 - y = 14$ and $25+ y = 36 $ and $14$ isn't square.

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