If $n=2^3 \cdot 3^2 \cdot 5$, how many even positive factors does $n$ have?
I know that to find the number of factors of a number, I need to use the prime factorization and add one to each exponent, then multiply them together. For example, if $m=2^4 \cdot 3^2 \cdot 5$, then $m$ has $(4+1)(2+1)(1+1) = 30$ factors. But this includes both even and odd factors. To find only the even factors, I need to make sure that the factor has a factor of 2 in it. That means I can fix the exponent of 2 to be any positive number from 1 to 3, and vary the exponents of 3 and 5 freely. So the number of even factors is $3 \cdot (2+1)(1+1) = 18$. 