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s(x) d(x) = 96, sum and no of positive divisors

Source: Austria Beginners' Competition 2018 p4

February 24, 2020
number theorysum of divisorsnumber of divisors

Problem Statement

For a positive integer nn we denote by d(n)d(n) the number of positive divisors of nn and by s(n)s(n) the sum of these divisors. For example, d(2018)d(2018) is equal to 44 since 20182018 has four divisors (1,2,1009,2018)(1, 2, 1009, 2018) and s(2018)=1+2+1009+2018=3030s(2018) = 1 + 2 + 1009 + 2018 = 3030. Determine all positive integers xx such that s(x)ā‹…d(x)=96s(x) \cdot d(x) = 96.
(Richard Henner)
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