<s>[INST] if a,b,c,d are integers such that 5 divides ($a^4+b^4+c^4+d^4$) then 5 also divides (a+b+c+d) i want to show that
if a,b,c,d are integers such that 5 divides ($a^4+b^4+c^4+d^4$) then 5 also divides (a+b+c+d) but im not quite sure how to aproach this problem.
 [/INST] Hint
$$a^4-1=(a^2-1)(a^2+1)=(a-1)(a+1)(a^2-4+5)=(a-2)(a-1)(a+1)(a+2)+5(a^2-1)$$
Now, if $5 \nmid a$ one of $(a-2),(a-1),(a+1),(a+2)$ is divisible by $5$.
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