Consider a prime $p$ of the form $4k+3$. Show that for any pair of integers $(a,b)$ such that p does not divide the gcd of $a$ and $b$, there exists $(k,l)∈ Z$ such that $ak-bl \equiv1 \pmod p$ and $al \equiv -bk \pmod p$.
My attempt:
Using the hint, I reduced the problem as follows-
"Prove that there exists some $r$ such that $r(a^2+b^2) \equiv 1 \pmod p$"
Now if I take $r=(a^2+b^2)^{p-2}$, then we get,
$$(a^2+b^2)^{p-1} \equiv 1 \pmod p$$
and thats Euler's theorem. Only I need to prove that $$(a^2+b^2)^{p-1} \equiv 0 \pmod p$$ can not be true.