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Fractions divisible by a fixed integer

Source: Poland 73-3-2

March 31, 2022
number theory

Problem Statement

Let m,n≥2m,n\ge 2 be given integers. Prove that there exist positive integers a1<a2<…<ama_1<a_2<\ldots<a_m so that for any 1≤i<j≤m1\le i<j\le m the number ajaj−ai\frac{a_j}{a_j-a_i} is an integer divisible by nn.
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