Let p be an odd prime. Show that 1^p + 2^p + 3^p + .... + (p-1)^p == 0 (mod p)

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Mathematics

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Elementary Number Theory

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Explanation & Answer

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)

=p(p+1)/2, 

since p is odd prime, (p+1)/2 is an integer, thus 

p((p+1)/2 ==0 mod p. 


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