Question

Find number of divisors of $2^{10}\times 3^{15}\times {15}^{13}$

Answer 1

Let $d\left(n\right)$ be the number of divisors for the natural number $n$, where is repesented as the products of prime numbers as $n=p^a{q}^b{r}^c$ , where $p,q,r$ are prime numbers. Then, the number of divisors:-
$d\left(n\right)=(a+1)(b+1)(c+1)$
Now,
$d(2^{10}\times 3^{15}\times {15}^{13})=d(2^{10}\times 3^{15}\times 3^{13}\times 5^{13})$
$=d(2^{10}\times 3^{28}\times 5^{13})$
$=(10+1)(28+1)(13+1)$
$=11\times 29\times 14$
$=4466$
Thus, the number of divisors are $4466$