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exactly \sqrt[2010]{n} positive integers x \le n : p^{2010}|x^p-1, q^{2010}|x^-1

Source: 2010 Indonesia TST stage 2 test 5 p4

December 16, 2020
number theory

Problem Statement

Let nn be a positive integer with n=p2010q2010n = p^{2010}q^{2010} for two odd primes pp and qq. Show that there exist exactly n2010\sqrt[2010]{n} positive integers x≤nx \le n such that p2010∣xp−1p^{2010}|x^p - 1 and q2010∣xq−1q^{2010}|x^q - 1.
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