How many different positive integer factors does have? How many different positive integer factors does $(2^7)(3^4)(7^3)(23^5)$ have?
Do we have to do any combinations between the powers here? 

Note the following:


*

*$\gcd(2,3)=1$

*$\gcd(2,7)=1$

*$\gcd(2,23)=1$

*$\gcd(3,7)=1$

*$\gcd(3,23)=1$

*$\gcd(7,23)=1$


Therefore, the number of different positive integer divisors of $(2^7)(3^4)(7^3)(23^5)$ is:
$$(7+1)\cdot(4+1)\cdot(3+1)\cdot(5+1)=960$$
