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Find the residue of $p^N+q^N \mod 6\cdot 12^n$.

Source: 39-th Vietnamese Mathematical Olympiad 2001

March 19, 2007

number theoryrelatively primenumber theory proposed

Problem Statement

Let

N = 6^{n}

, where

n

is a positive integer, and let

M = a^{N}+b^{N}

, where

a

and

b

are relatively prime integers greater than

1. M

has at least two odd divisors greater than

1

are

p,q

. Find the residue of

p^{N}+q^{N}\mod 6\cdot 12^{n}