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Integer a_k such that b - a^n_k is divisible by k

Source: IMO Shortlist 2007, N2, Ukrainian TST 2008 Problem 10

July 13, 2008

modular arithmeticnumber theoryDivisibilityIMO ShortlistHi

Problem Statement

Let

b,n > 1

be integers. Suppose that for each

k > 1

there exists an integer

a_k

such that

b - a^n_k

is divisible by

k

. Prove that

b = A^n

for some integer

A

.
Author: Dan Brown, Canada