Is there a relation between prime factors and total factors?

• Dec 7th 2009, 09:01 AM
dave1022
Is there a relation between prime factors and total factors?
For example, the prime factors of 28 are 2, 2, 7 and the factors are 1, 2, 4, 7, 14, 28

I was just wondering if there was a relationship between the number of prime factors and the number of total factors?

(I figured this would have something to do with combos or perms)

Regards,
David
• Dec 7th 2009, 10:42 AM
Plato
Quote:

Originally Posted by dave1022
For example, the prime factors of 28 are 2, 2, 7 and the factors are 1, 2, 4, 7, 14, 28
I was just wondering if there was a relationship between the number of prime factors and the number of total factors?

(I figured this would have something to do with combos or perms)

It has nothing to do with permutations/combinations, but it is a counting problem.

Written in prime factorization form: $28=2^2\cdot 7^1$.
Even though it is not the custom to use 1 as an exponent, I included it to make a point.
Add one to each exponent and multiply, $(2+1)(1+1)=6$.
So $28$ has six factors. We add the one to account for using 0 as an exponent.

Here is another example: $720=2^4\cdot 3^2\cdot 5^1$.
Thus $720$ has $(4+1)(2+1)(1+1)=30$ factors.