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divisibility with condition on lcm

Source: Baltic Way 2021, Problem 20

November 15, 2021
number theorynumber theory proposedDivisibilityleast common multiple

Problem Statement

Let n≥2n\ge 2 be an integer. Given numbers a1,a2,…,an∈{1,2,3,…,2n}a_1, a_2, \ldots, a_n \in \{1,2,3,\ldots,2n\} such that lcm⁡(ai,aj)>2n\operatorname{lcm}(a_i,a_j)>2n for all 1≤i<j≤n1\le i<j\le n, prove that a1a2…an∣(n+1)(n+2)…(2n−1)(2n).a_1a_2\ldots a_n \mid (n+1)(n+2)\ldots (2n-1)(2n).
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