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Divisibility on a sequence

Source: Kürschák competition 2011 P1

March 2, 2020
number theoryNumber sequence

Problem Statement

Let a1,a2,...a_1, a_2,... be an infinite sequence of positive integers such that for any k,ℓ∈Z+k,\ell\in \mathbb{Z_+}, ak+ℓa_{k+\ell} is divisible by gcd⁡(ak,aℓ)\gcd(a_k,a_\ell). Prove that for any integers 1⩽k⩽n1\leqslant k\leqslant n, anan−1…an−k+1a_na_{n-1}\dots a_{n-k+1} is divisible by akak−1…a1a_ka_{k-1}\dots a_1.
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