Question

If $n\ge 1$ is any integer, $d\left(n\right)$ denotes the number of positive factors of $n$, then for any prime number $p,d\left(d\right(d\left(p^7\right)\left)\right)=$

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is C $3$

For a number being $p^n$ the total number of positive factors excluding $1$ and the number itself will be $n-1$.
Therefore total number of positive factors will be $n-1+2$
$=n+1$
Hence for $p^7$ we will have $8$ factors.
Now for $8=2^3$.
Hence $8$ will have $4$ factors.
Now $4=2^2$, hence the number of factors will be $3$.