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 No Subject by Charlie)

I don't know why in the previous post I started by raising the higher primes to the 4th power and the lower primes to the first, rather than the other way around.

The lowest number with 1,000,000 factors (part C) is 37 * 31 * 29 * 23 * 19 * 17 * 13^4 * 11^4 * 7^4 * 5^4 * 3^4 * 2^4. Nothing is gained from combining factors here. The million factors are 37^a * 31^b * 29^c * 23^d * 19^e * 17^f * 13^g * 11^h * 7^i * 5^j * 3^k * 2^l, where a through f are each 0 or 1, and g through l are each zero through 4. It evaluates to about 2.00961610708939 * 10^26, or exactly 200,961,610,708,938,459,249,870,000.

That's part C.  For part B, with 1,000 factors, it comes out to

13 * 11 * 7 * 5^4 * 3^4 * 2^4 = 810,810,000.

 

Posted by Charlie on 2005-07-08 19:26:44