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Increasing sequence with gcd property!

Source: India TST 2001 Day 5 Problem 2

January 31, 2015
number theorygreatest common divisornumber theory unsolved

Problem Statement

A strictly increasing sequence (an)(a_n) has the property that gcd⁡(am,an)=agcd⁡(m,n)\gcd(a_m,a_n) = a_{\gcd(m,n)} for all m,n∈Nm,n\in \mathbb{N}. Suppose kk is the least positive integer for which there exist positive integers r<k<sr < k < s such that ak2=arasa_k^2 = a_ra_s. Prove that r∣kr | k and k∣sk | s.
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