2 x 3 x 5 = 30
How about: 210
30
3
2*3*5 = 30
16 if the prime factors don't have to be distinct. 210 if they do.
Seven distinct factors.
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
240 is the smallest number with twenty factors.
11
The smallest number greater than 200 that has exactly 4 prime factors is 210. This number can be expressed as the product of its prime factors: (2 \times 3 \times 5 \times 7), which indeed yields four distinct prime factors. Other combinations, such as using powers of primes, do not yield a smaller number exceeding 200.
24edit: nonethe smallest number with 12 distinct factors would be2*3*5*7*11*13*17*19*23*29*31*37>>100