just to verify whether the below method is a fluke or correct in approach:

even factors of 135000 and 108000 would not vary for powers of 2 and 3 as in both the cases for 135000, 2 and 3 powers are less or equal to that of for 108000. only 5 will make the difference. For 135000 power of 5 can take values 0,1,2,3 and 4 but for 108000 power of 5 can take values 0,1 and 2 only.

So total number of factors = 3*4*2 = 24

3 => 1,2,3 powers of 2

4=> 0,1,2,3 powers of 3

2 => 3,4 powers of 5