Number Theory; Difficulty: Hard

I would recommend Rich's approach: test numbers and try to disprove the Roman Numeral statements.

However, there is a larger theory here:
Whenever you multiply a given number by a prime (such as 13), you always double the number of distinct factors... unless that original number already contained that particular prime as a factor.

If x = 6, x has factors of 1, 2, 3, and 6. If we multiply it by 13, then 13x will still contain all of those factors, but it will also contain 13 times each of those factors: (1)(13) = 13, (2)(13) = 26, (3)(13) = 39, and (6)(13) = 78. So we've doubled the number of factors.

The only exception is if x already contained a factor of 13. For example, if x = 26, it will have factors of 1, 2, 13, and 26. 13x will have factors of 1, 2, 13, 26, 169, and 338. (1)(13) and (2)(13) were already included in our original count, and we don't double-count factors.

Therefore, if multiplying x by a prime doubles the number of factors, x must not have contained that prime as a factor.