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equations with product of positive divisors P(n) = 15n, P(n) = 15n^2

Source: Dutch NMO 2013 p4

September 6, 2019
number theoryequationProductpositiveDivisorsdivisor

Problem Statement

For a positive integer n the number P(n)P(n) is the product of the positive divisors of nn. For example, P(20)=8000P(20) = 8000, as the positive divisors of 2020 are 1,2,4,5,101, 2, 4, 5, 10 and 2020, whose product is 1⋅2⋅4⋅5⋅10⋅20=80001 \cdot 2 \cdot 4 \cdot 5 \cdot 10 \cdot 20 = 8000. (a) Find all positive integers nn satisfying P(n)=15nP(n) = 15n. (b) Show that there exists no positive integer nn such that P(n)=15n2P(n) = 15n^2.
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