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n^5 +5^n is divisible by 11 iff n^5 x 5^n +1 is divisible by 11

Source: 2024 Austrian Regional Competition For Advanced Students p4

March 26, 2024
number theorydivisibledivides

Problem Statement

Let nn be a positive integer. Prove that a(n)=n5+5na(n) = n^5 +5^n is divisible by 1111 if and only if b(n)=n5⋅5n+1b(n) = n^5 · 5^n +1 is divisible by 1111.
(Walther Janous)
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