Thaaaaaank you very much for your help. I will show you my solution:
So we use induction on the number of prime numbers appears in
;
If we have only one prime, then
and the divisors of
are
. Hence the number of divisors of
is
.
Inductive Step: Suppose that
contains
prime numbers, and the statement is true for
. i.e
and the number of divisors of
is
Our goal is to show that the statement is true for
which contain
prime numbers.
We have
. Hence
By inductive hypothesis, the number of divisors of
is
and we know that the number of divisors of
is
. So, the number of divisors of
is
which is
This is my solution. If there is any mistakes or comments please guide me. Thank you again