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equal periodic sequences

Source: 7th QEDMO problem 11 (14. - 17. 1. 2010) https://artofproblemsolving.com/community/c1512515_qedmo_200507

May 9, 2021
algebraperiodic

Problem Statement

Let mm and nn be two natural numbers and let d=gcd(m,n)d = gcd (m, n) their greatest common divisor. Let a1,a2,...a_1, a_2,... and b1,b2,...b_1, b_2, ... be two sequences of integers which are periodic with periods mm and nn respectively (this means that ai+m=aia_{i + m} = a_i and bi+n=bib_{i + n} = b_i for all natural numbers i≥1i \ge 1, note that there could be smaller periods). Prove that if the two sequences on the first m+n−dm + n - d terms match (i.e. ai=bia_i = b_i for all i∈{1,2,...,m+n−d}i \in \{1, 2, ..., m + n - d\}), then they are the same (so ai=bia_i = b_i for all natural i≥1i \ge 1).
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