Is there a relation between prime factors and total factors?

Oct 2009
5
0
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
 
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Aug 2006
22,432
8,612
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: \(\displaystyle 28=2^2\cdot 7^1\).
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, \(\displaystyle (2+1)(1+1)=6\).
So \(\displaystyle 28\) has six factors. We add the one to account for using 0 as an exponent.

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