<s>[INST] $5$ divides one of $x,y,z$ in $x^2+y^2=z^2$? 1) If $x$, $y$, and $z$ are positive integers for which $gcd(x,y,z) =1$ and $x^2+y^2=z^2$, show that $3$$∣$$xy$.
2) Now again if $x$, $y$, and $z$ are positive integers for which $gcd(x,y,z) =1$ and $x^2+y^2=z^2$, can a similar result to the one in (part 1) be said mod $5$? Not that $5$$∣$$xy$, but that $5$ divides one of $x,y,z$?
I believe I have 1) down: 
We note that $3$ is a prime, so $3∣xy⟺3∣x$ or $3∣y$. Suppose to the contrary that $3∤x$, and $3∤y$, then:$x=1$, $2$ mod $3$ and $y=1$, $2$ mod $3$. Then:
$x^2+y^2=2$ mod $3$, and $z^2=0$, $1$ mod $3$. Contradiction, proving the claim.
But I am having trouble with 2). Can anyone show a solution for 2)?
 [/INST] Squares are $0,1$ or $-1\bmod 5$
If both $x$ and $y$ are not multiples of $5$ then $x^2\equiv \pm 1$ and $y^2\equiv \pm 1\bmod 5$
This implies $x^2+y^2\equiv -2,0$ or $2\bmod 5$.
Of course, since $x^2+y^2$ is a square it must be $0\bmod 5$ given the previous options.
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