Question

Number of distinct prime factors of $2^{10}\times 3^{15}\times {15}^{13}$

Answer 1

In number theory, the number of distinct prime factors of an integer $n$ is denoted by $\omega \left(n\right)$. Also,
$\omega \left(mn\right)=\omega \left(m\right)+\omega \left(n\right)$ and for any positive integers $n$ and $x$, $\omega \left(n^x\right)=\omega \left(n\right)$
Now,
$\omega (2^{10}\times 3^{15}\times {15}^{13})$
$=\omega (2^{10}\times 3^{15}\times 3^{13}\times 5^{13})$
$=\omega (2^{10}\times 3^{28}\times 5^{13})$
$=\omega \left(2^{10}\right)+\omega \left(3^{28}\right)+\omega \left(5^{13}\right)$
$=\omega \left(2\right)+\omega \left(3\right)+\omega \left(5\right)$
$=1+1+1$
$=3$