Question

If $p,q$ are two distinct primes, then $\sigma \left(pq\right)$ equals (where the operation $\sigma$ on a number $n$ is defined as the sum of all divisors of the number $n$.)

• A. $\sigma \left(p\right)+\sigma \left(q\right)$
• B. $\sigma \left(p\right)\sigma \left(q\right)$
• C. $q\sigma \left(p\right)+p\sigma \left(q\right)$
• D. None of these

Answer 1

The correct option is B $\sigma \left(p\right)\sigma \left(q\right)$

If $p$ and $q$ are distinct prime numbers then divisors of $p$ are $1$ and $p$ itself.

Similarly divisors of $q$ are $1$ and $q$ itself.
Hence,
$\sigma \left(p\right)=1+p$ and $\sigma \left(q\right)=1+q$
Since, $p$ and $q$ are distinct primes, the factors of $pq$ will be $1,p,q$ and $pq$.
$\therefore \sigma \left(pq\right)=1+p+q+pq$
$=1+p+q(1+p)$
$=(1+p)(1+q)$
$=\sigma \left(p\right)\sigma \left(q\right)$

Hence, $\sigma \left(pq\right)=\sigma \left(p\right)\sigma \left(q\right)$