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Finding Possible Values in a Remainder Question

Consider the following question: 

N is a positive integer. When N is divided by 13, the remainder is 5. When N is divided by 46, the remainder is 31. What is the smallest possible value of N?

Before we examine the solution to this question, I’d like to set up this article by asking an easier question: 

N is a positive integer. When N is divided by 7, the remainder is 3. What are three possible values of N?

Does your list include 3 as a possible value for N? It should since 3 is the smallest number that meets the given criteria (3 divided by 7 equals 0 with remainder 3).

When it comes to GMAT questions involving remainders, it’s often useful to begin listing numbers that meet the given criteria. When it comes to listing possible values, we have a useful rule:

If N and D are positive integers, and if N divided by D is equal to Q with remainder R, then the possible values of N are: R, R+D, R+2D, R+3D,. . . 

Example: When positive integer W is divided by 6, the remainder is 5. Given this information, the possible values of W are: 5, 5+6, 5+2(6), 5+3(6), 5+4(6), . . .

Upon simplification, we find that the possible values of W are: 5, 11, 17, 23, 29, . . .

The important takeaway in this article is that you can sometimes save yourself a lot of work by listing possible values and, more importantly, by including the smallest possible value in that list.

Now back to the original question:

N is a positive integer. When N is divided by 13, the remainder is 5. When N is divided by 46, the remainder is 31. What is the smallest possible value of N?

First, if N is divided by 13 gives us remainder 5, the possible values of N are: 5, 18, 31, 44, 57, ...

Second, if N is divided by 46 gives us remainder 31, the possible values of N are: 31, ... STOP.

Since both lists include 31, the answer to our original question must be 31.

So, when you encounter a GMAT remainder question, one approach may involve identifying possible values of a certain number. When identifying those values, it may be to your advantage to identify the smallest possible value.