# Counting Principle

1. Jan 23, 2010

### weiji

How many divisors does 55,125 have? For example, 55,125 = (3)^2 . (5)^3 . (7)^2

2. Jan 23, 2010

### tiny-tim

Welcome to PF!

Hi weiji! Welcome to PF!

(try using the X2 tag just above the Reply box )
Well, each divisor has to be of the form 3a5b7c, wiht a b and c integers > 0 …

so how many is that?

3. Jan 23, 2010

### weiji

I did a very long calculation by assume a=1, b=1 ; a=1,c=1 ; b=1,c=1, from here, I know 1575x35 = 2625x21 = 3675x15. Then I calculate each possible answer, I got 36 divisors. But is there any faster way? I really have no idea. :(

4. Jan 23, 2010

### JSuarez

If the prime factorization of $$n$$ is $$n=p_1^{e_1}p_2^{e_2}...p_n^{e_n}$$. Now, in any divisor, each prime factor's exponent $$a$$ range from $$0\leq a \leq e_i$$.

5. Jan 24, 2010

### weiji

Thanks for sharing. By the way, I'm new here and nice to meet you all.