Let be the number of positive divisors of . Let be the number of positive divisors of which have remainders when divided by . Find all positive integral values of the fraction .
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 $\frac{\tau(10n)}{\tau_1(10n)}$.