<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] Remember Eulers theorem applied to the prime $5$ : $$ a^4 = 1 \mod 5 \iff a \mod 5 \neq 0$$ so we have $$(a^4+b^4+c^4+d^4) \mod 5 = 0 \iff  a=b=c=d=0 \mod 5$$
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