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Divides lcm of any n pairs

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

August 29, 2024

number theoryleast common multipleIndonesiaIndonesia MOLCM

Problem Statement

Let

n \ge 2

be a positive integer. Suppose

a_1, a_2, \dots, a_n

are distinct integers. For

k = 1, 2, \dots, n

, let

s_k := \prod_{\substack{i \not= k, \\ 1 \le i \le n}} |a_k - a_i|,

i.e.

s_k

is the product of all terms of the form

|a_k - a_i|

, where

i \in \{ 1, 2, \dots, n \}

and

i \not= k

. Find the largest positive integer

M

such that

M

divides the least common multiple of

s_1, s_2, \dots, s_n

for any choices of

a_1, a_2, \dots, a_n

.

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