Elementary Number Theory
From Fermat little theorem we have k^p = k (mod p), k is an integer and p is a prime. therefore,
1^p+2^p+....+(p-1)^p = 1+2+3+....+(p-1) (mod p)
since p is odd prime, (p+1)/2 is an integer, thus
p((p+1)/2 ==0 mod p.
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