That would be 503 according to me. Regardless of whether the factors are distinct or not. Sorry, checked by computer, no real maths here.
1) How many distinct prime factors in the prime factorisation?
2) How many divisors?
3) How many not-necessarily-distinct prime factors in the prime factorisation?
Personally I would like the question to be more specific, but others may claim it is clearly one of these three and not the other two.
Edit: and a fourth
4) How many proper divisors?
(and apparently "d" comes before "n".... chalk it up to being groggy in the morning)
I agree with Undefined, "factor" is just too close to "prime factor". And it seemed more interestingYou're right, I'm just more used to seeing the word divisor used. I also interpreted the question as asking how many divisors upon my first reading but didn't want to discredit Bacterius's answer. But if the question wants prime factors, it should say prime factors.
Going by the same token, I don't think that understanding of factor is "modernly" accurate, based on my reading over the past few months. But I don't want to fight over words, we're on a maths forum and I guess either the OP wasn't clear enough, either I was just too stupid to hit the point.
If we consider 'three factors' as 'three distinct prime factors' [so that, for example, 1 is not prime number and is not a 'factor'...] a simple systematic search can be done...
... so that 14 different numers less than 2010 expressed as product of three distinct prime fators have been found till now. But we can proceed futher...
... and so on...