I was wondering if there is a quick way to count the number of factors that a particular number has. I have run into two problems so far where I have needed to do this and was wondering if there is a shortcut. For example:

OG PS # 209

How many different postive integers are factors of 441:

So by adding the digits we can see that 4+4+1 = 9 so 441 is divisible by 9, 441/9 = 49
So we can see that the prime factors of 441 are 7, 7, 3, 3

My question is is there an easy way to figure out how many different factors you can come up with by combining the prime factors:
We know four factors off the bat: 1, 3,7, 441
Then we have to calculate the rest:
3*3 = 9
3*7 = 21
7*7 = 49
3*3*7 = 56
7*7*3 = 147

So that comes to a total of 9 factors. I was wondering if there is a way to figure out (possibly using combinatorics) how to calculate the number of factors you can obtain from 3,3,7,7. If you can calculate this number quickly you know there are just two more factors 1 and the number 441 itself.

Thanks

ack

Excellent question - and, actually, the shortcut is essentially laid out below - you just didn't use it. You don't actually have to finish the multiplication. As long as the factors you multiply are different, the product will be unique - so don't finish the math. Just write out 7, 7, 3, 3 and count the number of different combos you've got. :)