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Integral ratio of divisors to divisors 1 mod 3 of 10n

Source: 2016 IMO Shortlist N2

July 19, 2017

number theoryIMO Shortlist

Problem Statement

Let

\tau(n)

be the number of positive divisors of

n

. Let

\tau_1(n)

be the number of positive divisors of

n

which have remainders

1

when divided by

3

. Find all positive integral values of the fraction

\frac{\tau(10n)}{\tau_1(10n)}