Why must 12N/5 be divisible by 12? I understand that the total population of the school would be 12N/5. I also understand that because the question involves people, the answer must be a whole number. However, for the answer to be a whole number, it is my understanding that N must be divisible by 5. If 12N/5 were divided by 12 we would still end up with N/5.
We see that N must be divisible by 5, but the question doesn't ask about N. The question asks about 12N/5. Well, clearly, 12N/5 must be divisible by 12, so this question really boils down to "which of these five numbers is divisible by 12"? That's the real question-inside-the-question, the actual mathematical task at hand.
So you're right that it's not a given that 12N/5 will be an integer—that's why we are looking for an answer choice that is divisible by 12. The equation that we're trying to solve, 12(N/5), is the total 2004 population. However if you'll recall, N is only equal to the 2003 girl OR boy population. So, we are not solving for x, per se, but we are solving for the total student population at Jefferson High in 2004 (12(N/5)). This is why we are not looking for a number that is divisible by 5, but a number that is divisible by 12. If we were looking for a number that was divisible by 5, we would get an answer only for N, which would be an answer to the 2003 girl or boy population, and not the total 2004 population (which is what the question is asking for).