Which number under 1000 has the most different prime factors?

Answer 1

It's relatively easy to show that there are no numbers less than 1000 that are made of more than 4 different prime factors. The 5 smallest primes are 2, 3, 5, 7 and 11, and 2x3x5x7x11=2310. If the smallest 5 different primes multiplied together gives a product greater than 1000, then any other combination of 5 different primes would yield an even bigger number. So any number less than 1000 that is comprised of different prime numbers must only have 4 prime factors. So the smallest number that's a product of 4 different primes is 210 (=2x3x5x7), but there are several other numbers less than 1000 that are made of 4 different prime factors. A complete list follows: 2x3x5x7 = 210 2x3x5x11 = 330 2x3x7x11 = 462 2x5x7x11 = 770 2x3x5x13 = 390 2x3x11x13 = 858 2x3x5x17 = 510 2x3x7x17 = 714 2x3x5x19 = 570 2x3x7x19 = 798 2x3x5x23 = 690 2x3x7x23 = 966 2x3x5x29 = 870 2x3x5x31 = 930