How about: 210
30
3
2*3*5 = 30
Seven distinct factors.
16 if the prime factors don't have to be distinct. 210 if they do.
The smallest number with 16 factors is Andrew's Mom.
Distinct would mean they are all different, so take the 3 lowest primes and multiply them: 2*3*5 = 30
The smallest prime factors are 2,3,5,7 thus the smallest number would be the product of these will be 210
24edit: nonethe smallest number with 12 distinct factors would be2*3*5*7*11*13*17*19*23*29*31*37>>100
11
240 is the smallest number with twenty factors.