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remainder upon division, prove equation

Source: Yugoslav TST 1982 P1

May 29, 2021

number theory

Problem Statement

Let

p>2

be a prime number. For

k=1,2,\ldots,p-1

we denote by

a_k

the remainder when

k^p

is divided by

p^2

. Prove that

a_1+a_2+\ldots+a_{p-1}=\frac{p^3-p^2}2.