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Difference of factors of integers

Source: China Mathematical Olympiad 2014 Q2

December 21, 2013
number theory proposednumber theory

Problem Statement

For the integer n>1n>1, define D(n)={a−b∣ab=n,a>b>0,a,b∈N}D(n)=\{ a-b\mid ab=n, a>b>0, a,b\in\mathbb{N} \}. Prove that for any integer k>1k>1, there exists pairwise distinct positive integers n1,n2,…,nkn_1,n_2,\ldots,n_k such that n1,…,nk>1n_1,\ldots,n_k>1 and ∣D(n1)∩D(n2)∩⋯∩D(nk)∣≥2|D(n_1)\cap D(n_2)\cap\cdots\cap D(n_k)|\geq 2.
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