Chick wrote:What is the remainder when the positive integer x is divided by 6?

(1) When x is divided by 2, the remainder is 1; and when x is divided by 3, the remainder is 0.

(2) When x is divided by 12, the remainder is 3.

Answer: D

(1) When x is divided by 2, the remainder is 1; and when x is divided by 3, the remainder is 0.

Because it is more concrete, let's begin by focussing on the second clause (after the semi-colon): when x is divided by 3, the remainder is 0. In other words, x is a multiple of 3. Write out the first 3 or 4 (positive) multiples of three: 3, 6, 9, 12.

Now, let's look at the first clause: when x is divided by 2, the remainder is 1. Well, 3 divided by 2 leaves a remainder of 1 while 6 divides 2 evenly. And 9 divided by 2 leaves a remainder of 1 while 12 divides 2 evenly. Therefore, x is an odd multiple of 3, like 3 or 9 (or 15 or 21, etc). Any odd multiple of 3 divided by 6 will leave a remainder of 3. For example, 3 divided by 6 leaves a remainder of 3; 9 divided by 6 leaves a remainder of 3; 15 divided by 6 leaves a remainder of 3; and, so on.

Sufficient.

(2) When x is divided by 12, the remainder is 3.

So, if "n" is a multiple of 12, then x is n + 3. For example, 12 is a multiple of 12; then x is 15. 15 divided by 12 leaves a remainder of 3, so the statement is satisfied. Or, 24 is a multiple of 12; then x is 27. 27 divided by 12 leaves a remainder of 3, and so the statement is satisfied. Again, x can be any multiple of 12 plus 3. So x can be 15 (12 +3); x can be 27 (24 +3); x can be 39 (36 + 3); and so on. 15 divided by 6 leaves a remainder of 3. 27 divided by 6 leaves a remainder of 3. 39 divided by 6 leaves a remainder of 3. And so on. (In fact, under statement two, x is every second odd multiple of 3. And, from analyzing statement one, we already know that any odd multiple of 3 will give the same remainder when divided by 6).

Sufficient.

Both statements independently sufficient; choose D.

In the future, please post DS questions in the DS (rather than the PS) forum. This keeps things organized around here!