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Number theory with powers of two

Source: Russian TST 2016, Day 10 P1 (Group A), P2 (Group B)

April 19, 2023
number theoryDivisibility

Problem Statement

Let aa{} and bb{} be natural numbers greater than one. Let nn{} be a natural number for which a∣2n−1a\mid 2^n-1 and b∣2n+1b\mid 2^n+1. Prove that there is no natural kk{} such that a∣2k+1a\mid 2^k+1 and b∣2k−1b\mid 2^k-1.
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