studentps2011 wrote:If X and Y are nonzero integers, what is the remainder when X is divided by Y?
(1) When X is divided by 2Y, the remainder is 4
(2) When X+Y is divided by Y, the remainder is 4
OA:B[/spoiler]
1) If X divided by 2Y leaves a remainder of 4, then X must be 4 more than a multiple of 2Y: X=2Y*N+4 for some integer, N. Dividing both sides of the equation by Y gives X/Y=2N+4/Y. In words, this says "X divided by Y is 2N, plus 4/Y. Now, if Y>4, the remainder would also be 4, but if Y<4, then Y will divide into the 4 on the end and leave a different remainder. For example, if X=40, Y=3, X divided by 2Y is 40 divided by 6, which is 6 R4. But 40 divided by 3 is 13 remainder 1. However, if X=40 and Y=6, then 40 divided by 12 is 3 R4, and 40 divided by 6 is 6 R4. In this case both remainders are 4. INSUFFICIENT.
2) X+Y divided by Y leaves a remainder of 4. X+Y=Y*M+4 for some integer M. Divide both sides by Y----> X/Y+1=M+4/Y---> X/Y=(M-1)+4/Y. This says that X divided by Y is M-1 + 4/Y. In this case, Y MUST be greater than 4, though (do you see why?), so Y cannot be divided into 4, so the remainder must also be 4.
That's all kind of abstract, so it might help to look at some examples. Think about how remainders work. For example, let's look at a bunch of consecutive integers and their remainder when divided by 7:
(Number, remainder when divided by 7):
7,0
8,1
9,2
10,3
11,4
12,5
13,6
14,0
15,1
16,2
17,3
18,4
19,5
20,6
21,0
22,1
23,2
24,3
The remainders just keep cycling from 0 through 6 forever. Now, pick any number, note its remainder, and add 7. 12 has remainder 5, 19 has remainder 5. 9 has remainder 2, 16 has remainder 2. So, any time you add 7 or any multiple of 7, you just skip to another number that is in the same position in the remainder cycle.
In this case, the same type of thing is happening. X has some remainder when divided by Y. If we add Y to X, it's just going to jump it to the next number that is in that same spot in the remainder cycle. So if X+Y has a remainder of 4 when divided by Y, X must also have a remainder of 4 when divided by Y.
Thanks again for taking time to explain it in such detail.
