If 4 is divided into the positive integer x, it leaves a remainder of 3. If 9 is divided into x, it leaves a remainder of 4. If y is a positive integer such that x + y is divisible by 36, what is the smallest possible value of y?
The wording of this question is awful. The GMAT would never talk about one number "dividing into" another. Further, when this question says "it leaves a remainder of 3", the word "it" refers back to "4", when it should actually refer back either to "x" or more correctly, to the division itself. This question would be better as a Sentence Correction question.
This is, mathematically, an exact copy of question 68 in the PS section of the official Quant Review book, but with different numbers and with unclear wording. The first sentence in the official version of this question is worded as follows, and this is the wording you'll typically see on the real GMAT whenever remainders are being discussed:
"When positive integer n is divided by 5, the remainder is 1".
So the question quoted above should be phrased as follows:When the positive integer x is divided by 4, the remainder is 3. When x is divided by 9, the remainder is 4. What is the smallest positive integer y for which x+y is divisible by 36?
There is no need to use algebra here. If we can find any value for x at all that gives the right remainders, we can use it to answer the question. If we make a list of numbers which give a remainder of 3 when divided by 4:
3, 7, 11, 15, 19, 23, 27, 31, 35, ...
and a list of numbers which give a remainder of 4 when divided by 9:
4, 13, 22, 31, ...
then any potential value of x needs to be in both of the lists above. We can see that x could be equal to 31, in which case y would be 5, so the answer is 5.
In general, using pure algebra in remainders questions can be awkward. It's very often best just to find a number which 'works' and to use that number to answer the question.
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