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8

Part of 2024 Indonesia MO

Problems(1)

Divides lcm of any n pairs

Source: Indonesia National Mathematics Olympiad (INAMO) 2024 Problem 8

8/29/2024
Let n≥2n \ge 2 be a positive integer. Suppose a1,a2,…,ana_1, a_2, \dots, a_n are distinct integers. For k=1,2,…,nk = 1, 2, \dots, n, let sk:=∏i≠k,1≤i≤n∣ak−ai∣, s_k := \prod_{\substack{i \not= k, \\ 1 \le i \le n}} |a_k - a_i|, i.e. sks_k is the product of all terms of the form ∣ak−ai∣|a_k - a_i|, where i∈{1,2,…,n}i \in \{ 1, 2, \dots, n \} and i≠ki \not= k. Find the largest positive integer MM such that MM divides the least common multiple of s1,s2,…,sns_1, s_2, \dots, s_n for any choices of a1,a2,…,ana_1, a_2, \dots, a_n.
number theoryleast common multipleIndonesiaIndonesia MOLCM