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Number of solutions is always a perfect square

Source: 2019 Baltic Way P16

November 18, 2019
number theory

Problem Statement

For a positive integer NN, let f(N)f(N) be the number of ordered pairs of positive integers (a,b)(a,b) such that the number aba+b\frac{ab}{a+b} is a divisor of NN. Prove that f(N)f(N) is always a perfect square.
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