Question

The number of odd proper divisors of $3^p·6^m·{21}^n$ is equal to$?$

Answer 1

Find the number of proper odd divisors of the given number:

A number $3^p·6^m·{21}^n$ is given.

Rewrite the given number as follows:

$$\begin{gathered} 3^p·6^m·{21}^n=3^p·{(3·2)}^m·{(3·7)}^n \\ \Rightarrow 3^p·6^m·{21}^n=2^m·3^{p+m+n}{·7}^n \end{gathered}$$

So, the number of odd divisors of the given number is equal to $\mathbf{(}\mathbf{p}\mathbf{+}\mathbf{m}\mathbf{+}\mathbf{n}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{(}\mathbf{n}\mathbf{+}\mathbf{1}\mathbf{)}$.

And the number of proper odd divisors of the given number is equal to $(p+m+n+1)(n+1)-1$.

Therefore, the number of proper odd divisors of $3^p·6^m·{21}^n$ is equal to $\mathbf{(}\mathbf{p}\mathbf{+}\mathbf{m}\mathbf{+}\mathbf{n}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{(}\mathbf{n}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{-}\mathbf{1}$.