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Selecting divisors of square-free integers

Source: Kyiv City MO 2022 Round 1, Problem 9.4

January 23, 2022
number theory

Problem Statement

Let's call integer square-free if it's not divisible by p2p^2 for any prime pp. You are given a square-free integer n>1n>1, which has exactly dd positive divisors. Find the largest number of its divisors that you can choose, such that a2+abāˆ’na^2 + ab - n isn't a square of an integer for any a,ba, b among chosen divisors.
(Proposed by Oleksii Masalitin)
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