<s>[INST] Show a,b,c congruences when $a^2+b^2=c^2$ i would like to know how to prove that if $a^2+b^2=c^2$ then 
a) $3|a$ or $3|b$ 
b) $5|a$ or $5|b$ or $5|c$ 
c) $4|a$ or $4|b$ 
Also that $a^2+b^2+c^2+1$ can't be divisible by 8
Thank you very much!
 [/INST] Hint: Take the equation $a^2+b^2=c^2$ modulo 3. Also, note that
$$0^2\equiv 0\bmod 3,\qquad 1^2\equiv 1\bmod 3,\qquad 2^2\equiv 1\bmod 3,$$
and since any integer is equivalent modulo 3 to either 0, 1, or 2, we see that any square number, when taken modulo 3, cannot be equivalent to 2.
Do the same for the other moduli; work out the various cases according to what values $a^2$, $b^2$, and $c^2$ might take modulo $n$, rule out the cases that are impossible.
Also, keep in mind that if a prime number $p$ divides a square number $r^2$, then it must divide $r$.
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