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modular congruence for sequence

Source: Brazil TST 2001 Test 2 P4

April 28, 2021
algebranumber theory

Problem Statement

Prove that for all integers n≥3n\ge3 there exists a set An={a1,a2,…,an}A_n=\{a_1,a_2,\ldots,a_n\} of nn distinct natural numbers such that, for each i=1,2,…,ni=1,2,\ldots,n, ∏1≤k≤nk≠iak≡1(modai).\prod_{\small{\begin{matrix}1\le k\le n\\k\ne i\end{matrix}}}a_k\equiv1\pmod{a_i}.
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