Question

The number of proper divisors of $2^p·6^q·{15}^r$ is equal to?

• A. $\left(p+q+1\right)(q+r+1)(r+1)$
• B. $\left(p+q+1\right)(q+r+1)(r+1)-2$
• C. $\left(p+q\right)(q+r)r-2$
• D. None of these

Answer 1

The correct option is B

$\left(p+q+1\right)(q+r+1)(r+1)-2$

Explanation for the correct option:

Find the number of proper divisors of the given number.

A number $2^p·6^q·{15}^r$ is given.

Rewrite the given number as follows:

$$\begin{gathered} 2^p·6^q·{15}^r=2^p·{\left(2·3\right)}^q·{\left(3·5\right)}^r \\ \Rightarrow 2^p·6^q·{15}^r=2^{p+q}·3^{q+r}·5^r \end{gathered}$$

So, the total number of divisors is $(p+q+1)(q+r+1)(r+1)$.

since, $1$ and the number itself is not included in proper divisors.

Therefore, the number of proper divisors is $(p+q+1)(q+r+1)(r+1)-2$.

Hence, option $B$ is the correct option.