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Distant n-tuples under modulo

Source: 2022 China TST, Test 3 P3

April 30, 2022
modular arithmeticalgebranumber theorycombinatoricsabstract algebra

Problem Statement

Given a positive integer n≥2n \ge 2. Find all nn-tuples of positive integers (a1,a2,…,an)(a_1,a_2,\ldots,a_n), such that 1<a1≤a2≤a3≤⋯≤an1<a_1 \le a_2 \le a_3 \le \cdots \le a_n, a1a_1 is odd, and (1) M=12n(a1−1)a2a3⋯anM=\frac{1}{2^n}(a_1-1)a_2 a_3 \cdots a_n is a positive integer; (2) One can pick nn-tuples of integers (ki,1,ki,2,…,ki,n)(k_{i,1},k_{i,2},\ldots,k_{i,n}) for i=1,2,…,Mi=1,2,\ldots,M such that for any 1≤i1<i2≤M1 \le i_1 <i_2 \le M, there exists j∈{1,2,…,n}j \in \{1,2,\ldots,n\} such that ki1,j−ki2,j≢0,±1(modaj)k_{i_1,j}-k_{i_2,j} \not\equiv 0, \pm 1 \pmod{a_j}.
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