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Find the greatest n that always divides the product

Source: Lusophon Mathematical Olympiad 2024 Day 1 Problem 2

July 26, 2024
number theory

Problem Statement

For each set of five integers S={a1,a2,a3,a4,a5}S= \{a_1, a_2, a_3, a_4, a_5\} , let PSP_S be the product of all differences between two of the elements, namely
PS=(a5−a1)(a4−a1)(a3−a1)(a2−a1)(a5−a2)(a4−a2)(a3−a2)(a5−a3)(a4−a3)(a5−a4)P_S=(a_5-a_1)(a_4-a_1)(a_3-a_1)(a_2-a_1)(a_5-a_2)(a_4-a_2)(a_3-a_2)(a_5-a_3)(a_4-a_3)(a_5-a_4)
Determine the greatest integer nn such that given any set SS of five integers, nn divides PSP_S.
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