GRE & GMAT Mathy Tutor: how many factors?

How many unique factors does 40320 have?

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The prime factorization of

$40320$ yields:

$40320 = 2^7*3^2*5^1*7^1$

so any factor of of 40320 can be written in this form:

$2^a*3^b*5^c*7^d$
where

$a$ ,

$b$ ,

$c$ , and

$d$ are all non-negative integers and:

$a<=7$

$b<=2$

$c<=1$

$d<=1$

so the number of options for

$a$ is 8 (including 0), and for

$b$ : 3,

$c$ : 2, and

$d$ : 2.
so to count all the permutations of factors, we take the product:

$8*3*2*2 = 96$

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