how many factors?
2012-05-01
How many unique factors does 40320 have?
[Spoiler Below]
Video:
Text:
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`
[Spoiler Below]
Video:
Text:
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`