Question 40
Let $\tau(n)$ be the number of positive divisors of $n$. Let $\tau_1(n)$ be the number of positive divisors of $n$ which give remainder $1$ when divided by $3$. Find all possible integral values of the fraction
$$
\frac{\tau(10n)}{\tau_1(10n)}.
$$
$(\text{IMO Shortlist 2016})$
$$
\frac{\tau(10n)}{\tau_1(10n)}.
$$
$(\text{IMO Shortlist 2016})$