IMO 2016 Shortlist N2

Let τ(n) be the number of positive divisors of n. Let τ1(n) be the number of positive divisors of n which have remainder...

IMO 2016 Shortlist N2

Category: Number Theory

Problem

Let τ(n) be the number of positive divisors of n. Let τ1(n) be the number of positive divisors of n which have remainders 1 when divided by 3. Find all possible integral values of the fraction τ(10n) τ1(10n) .