PROBLEM NO.51 (B0MC-2,DAY-5)

[SANZEED]

Determine all positive integers $k$ such that
\[\frac{\tau (n^{2})}{\tau (n)}=k\]
for some $n$

[SANZEED]

Hint:

$k$ must be odd.

A general solution can be derived using the fact that every natural number can be written in the following form:\[2^{k}\times (2m+1)\]for some integer $k,m$.

[sm.joty]

may be $n=k=1$

[SANZEED]

No, for all odd $n$,\[\tau (n)\] divides \[\tau (n^{2})\],

note that \[\tau (n)\] means the number of common divisors of $n$ including $1,n$.