josephgmat800 wrote:
This is how I see this problem, could anybody point out where I'm wrong?
For maximum total numbers of prime factors for M while having 20 factors: 20 factors all being prime factors
If a number has 20 different prime factors, then the number will have a lot more than 20 factors in total. In fact, such a number would have at least 2^20 divisors in total, so would have more than a million divisors.
You could look at a simpler example - say a number has three prime divisors, so perhaps the number is (2)(3)(5) = 30. That number has eight divisors in total: 1, 2, 3, 5, 6, 10, 15, and 30. Because any combination of our prime divisors is also a divisor of 30, we have several divisors that are not prime. So that's the reason the number in this question cannot have nearly as many as twenty different prime factors.
There's a method you can learn that lets you count any number's divisors once you have that number's prime factorization. The solutions above are all using that method. If you haven't encountered that before, then I wouldn't start with the question posted here, which tests the concept in a much harder way than most questions do. I'd suggest you learn the theory first, from a good prep book, and then look at some easier questions before trying this one (and you could probably even skip this one, because I don't think you'll see something like it on the GMAT).
_________________
GMAT Tutor in Montreal
If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com