The product of 3305 and the 1-digit integer x is a 5-digit integer.

Wow. There are a lot of steps to this one. The key is going to be extremely careful organization (you should pick up on that when you see how many numbers there are in the answer choices, and the complexity of the question they're asking you.)

First, I read the whole thing. Then I approached the first sentence:

3305x is a 5-digit integer.

That probably limits the possible values for x. For instance, x can't be 1, 2, or 3, because the product would be too small. Is there an upper limit to the values of x? For instance, could x be 9? To check, I multiplied 3305*9 on my paper and found that the result had 5 digits. So, the acceptable values for x will be 4, 5, 6, 7, 8, and 9 (so far). I jotted those down on my scratch paper (large & clear).

4 5 6 7 8 9

The problem also says that the units digit of 3305x is 5. That means x must be odd. If x was even, the units digit would come out to 0. So, I crossed off all of the even numbers:

4 5 6 7 8 9

Now there are only three possible values for x. I don't see anything that limits it further than that, so I write down on my paper:

A = {5, 7, 9}

At this point, I also jot down the answer choices, A B C D E, and cross off everything except for D and E.

To figure out the values of y, we need to find the hundreds digit of 3305x. There's a rule that says that if you only want a particular digit, you can ignore everything past that digit. We can ignore the first 3 in 3305 and just treat it like 305, since it won't affect the hundreds digit of the result. That'll speed up the multiplication.

305 * 5 = 1525, hundreds digit = 5
305 * 7 = 2135, hundreds digit = 1
305 * 9 = 2745, hundreds digit = 7

So now I know B = {1, 5, 7}.

Finally, check this against the two remaining answer choices. D is correct.