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infinite sequence of positive integers

Source: 2023 HMIC P5

April 25, 2023

HMIC

Problem Statement

Let

a_1, a_2, \dots

be an infinite sequence of positive integers such that, for all positive integers

m

and

n,

we have that

a_{m+n}

divides

a_ma_n-1.

Prove that there exists an integer

C

such that, for all positive integers

k>C,

we have

a_k=1.