A. What is the lowest number that has exactly 10 distinct positive factors?
B. Exactly 1,000 distinct positive factors?
C. Exactly 1,000,000 distinct positive factors?
Example: The distinct factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. Thus 72 has 12 distinct factors.
(In reply to
general(ish) sol'n (w/o computer) by Josh70679)
After reading charlie's solution, i realized i miscounted my primes (*gasp*), and my answer for C is wrong (I left out 37 as a factor). This adds a wrinkle, as 2^5 = 32 < 37. That means factoring 1000000 into 10*5^5*2^5 instead of 5^6*2^6 is actually an improvement. The result using all prime factors is A (below) and the factoring using a 10 is B.
- A = 2^4 * 3^4 * 5^4 * 7^4 * 11^4 * 13^4 * 17 * 19 * 23 * 29 * 31 * 37
- B = 2^9 * 3^4 * 5^4 * 7^4 * 11^4 * 13^4 * 17 * 19 * 23 * 29 * 31
this means A/B = 37/(2^5) > 0, so B is actually better. I'm pretty sure it's now optimal, but I'm not going to spend the time to prove it ;).
PS:
- B = 173,804,636,288,811,640,432,320,000
- A = 200,961,610,708,938,459,249,870,000 (thanks Charlie)
re: general(ish) sol'n (w/o computer) -correction
