<s>[INST] $l^2+m^2=n^2$ $\implies$ $lm$ is always a multiple of 3 when $l,m,n,$ are positive integers. Let $l,m,n$ be any three positive integers such that $l^2+m^2=n^2$ 
Then prove that $lm$ is always a multiple of 3.
 [/INST] We show that $3$ must divde $lm$, by showing that $3$ divides $l$ or $3$ divides $m$. 
Any integer $x$ is either divisible by $3$, or is congruent to $1$ or $-1$ modulo $3$. And iff $x\equiv \pm 1\pmod{3}$, then $x^2\equiv 1 \pmod{3}$. 
Thus if neither $l$ nor $m$ is divisible by $3$, then $l^2+m^2\equiv 2\pmod{3}$. It follows that $l^2+m^2$ cannot be a perfect square. 
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