Factors of Factorials

Factorials have a lot of factors!

For example, $8!$ has the obvious factors of $8$, $7$, $6$, $5$, $4$, $3$, $2$, and $1$, but it has so many more!

$56$ is also a factor of $8!$ because $8 \times 7 = 56$.

$30$ is also a factor of $8!$ because $6 \times 5 = 30$.

Even $1$,$440$ is a factor of $8!$ because $8 \times 6 \times 5 \times 3 \times 2 = 1$,$440$. 

In fact, if you convert $8!$ into its prime factorization form, $2^7 \times 3^2 \times 5^1 \times 7^1$, and you do the prime factorization trick to find the number of positive factors, you will see that $8!$ has $96$ positive factors. If you're not sure where this came from, don't worry -- it's coming later in your studies.

$20!$ has a mind-numblingly large $41$,$040$ positive factors!