Number Theory?

Question

Can anyone help with this? Determine the smallest and second smallest positive integers which have exactly 12 distinct, positive divisors. Thanks for your assistance.

Answer 1

The candidates can be narrowed down to:

2³ * 3² = 72 (This number has 4 * 3 = 12 factors.)
2³ * 5² = 200 (This number has 4 * 3 = 12 factors.)
2 * 2 * 3 * 5 = 60 (This number has 3 * 2 * 2 = 12 factors.)
2 * 2 * 3 * 7 = 84 (This number has 3 * 2 * 2 = 12 factors.)
2^5 * 3 = 96 (This number has 6 * 2 = 12 factors.)

Looks like the first two are 60 and 72.

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On how to determine how many divisors a number has:

First break the number into its prime factors. First example, 60 = 2² * 3 * 5. From the factorization, we can see that the factors of any divisor of 60 has to have as its factors 2^x * 3^y * 5^z, where x = 0, 1, or 2, y = 0 or 1, and z = 0 or 1. There are 3 choices for x and 2 for y and z, so the total number of factors is 3 * 2 * 2 = 12. If you follow that, you'll have no trouble seeing how to generalize the logic.

Answer 2

It would have helped if you mentioned how much mathematics you already know. Modern number theory covers a lot of territory. A comprehensive book would be at least 6 feet thick. Your best bet is to go to the nearest college library and check out their number theory section. You may also want to check out their book store and see what text books the instructors are using.