A. What is the lowest base 12 positive integer that has exactly 10 (base 12) distinct positive factors?

B. Exactly 1,000 (base 12) distinct positive factors?

C. Exactly 1,000,000 (base 12) distinct positive factors?

For example, the distinct positive factors of 40 (base 12) are the base 12 numbers 1, 2, 3, 4, 6, 8, 10, 14, 20, and 40. Accordingly, 40 (base 12) has precisely A (base 12) distinct positive factors.

The number of factors an integer has is independent of the base in which it is written, so, for example, part A asks us to find the lowest positive integer with exactly 12 (the decimal equivalent of the base-12 number 10) divisors, and express that integer in base 12.

Part A (12 divisors):

12 = 2*2*3 = 4*3 = 2*6

so the number could be p1*p2*p3^2 or p1^3*p2^2 or p1*p2^5 or even p1^11.

In any given one of these choices, it would be best (for a lower number) to use 2 for the prime that's raised to the highest power, 3 for the next highest, etc.

2^2 * 3 * 5 = 60
2^3 * 3^2 = 72
2^5 * 3 = 96
2^11 = 2048

so the lowest is 60, which, expressed in base-12, is 50.

Posted by Charlie on 2010-10-11 11:48:57