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a_n is smallest pos integer such a_0 + a_1 +... + a_n is divisible by n

Source: 2019 Saudi Arabia IMO TST II p1

July 28, 2020
number theorydividesdivisible

Problem Statement

Let a0a_0 be an arbitrary positive integer. Let (an)(a_n) be infinite sequence of positive integers such that for every positive integer nn, the term ana_n is the smallest positive integer such that a0+a1+...+ana_0 + a_1 +... + a_n is divisible by nn. Prove that there exist NN such that an+1=ana_{n+1} = a_n for all n≥Nn \ge N
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