Not possible value?

Question

IF a natural number N has 12 factors , then which of the following is NOT a possible value for the number of factors of N^2 ?

(1)23

(2)33

(3)35

(4)55

(5)45

How to choose the correct answer ?

Answer 1

I'm not sure how to solve this analytically. However, I partially solved it by process of elimination. I wrote a program that found all numbers less than 1000 that have exactly 12 factors (including 1 and the number itself). That list of numbers is

60, 72, 84, 90, 96, 108, 126, 132, 140, 150
156, 160, 198, 200, 204, 220, 224, 228, 234, 260
276, 294, 306, 308, 315, 340, 342, 348, 350, 352
364, 372, 380, 392, 414, 416, 444, 460, 476, 486
490, 492, 495, 500, 516, 522, 525, 532, 544, 550
558, 564, 572, 580, 585, 608, 620, 636, 644, 650
666, 675, 693, 708, 726, 732, 735, 736, 738, 740
748, 765, 774, 804, 812, 819, 820, 825, 836, 846
850, 852, 855, 860, 868, 876, 884, 928, 940, 948
950, 954, 968, 975, 988, 992, 996

I figured this would be enough for a conclusive result. I then wrote another program that took each of these numbers in turn, squared it, then counted its factors. A printout of the number of factors of each of these numbers is

45, 33, 45, 35, 45, 45, 33, 45, 45, 45
45, 45, 45, 45, 45, 45, 45, 45, 45, 33
45, 45, 45, 35, 45, 33, 45, 45, 45, 33
45, 45, 45, 35, 45, 45, 45, 45, 33, 45
45, 45, 45, 45, 45, 33, 45, 45, 45, 45
45, 35, 45, 45, 45, 45, 45, 33, 45, 45
45, 45, 45, 45, 45, 45, 45, 45, 45, 45
45, 45, 45, 45, 45, 45, 45, 33, 45, 45
45, 45, 35, 45, 45, 33, 45

As you can see, 33, 35, and 45 appear in the list. However, neither 23 nor 55 appear, so the correct answer is one of these two.

Update:
After thinking about it some more, something occurred to me. Let's say N = 2048. In this case, N has a prime factorization that's just one number repeated (2048= 2^11). Therefore, the only numbers which are factors of 2048 are
2^0, 2^1, 2^2, 2^3, ... 2^11
which makes twelve factors in total.

Now,
2048^2 = 4194304 = 2^22
and we can apply similar logic to this number. The only factors are
2^0, 2^1, 2^2, 2^3, ... 2^22
which makes 23 factors in total.

Therefore, 55 (choice 4) is the only remaining possibility!

I don't know enough number theory to conclusively prove that N^2 can't ever have 55 factors, but I did prove that it is possible for N^2 to have 23, 33, 35, or 45 factors, therefore, by process of elimination, 55 is the correct choice.

Update 2:
I browsed some of your other questions. You ask some really tricky ones. Where do you get these questions?