# Listing Possible Values in a Remainder Question

## Remainders graphic.png

x and y are positive integers. When x is divided by 11, the remainder is 5, and when x is divided by 34, the remainder is 27. When y is divided by 17, the remainder is 11, and when y is divided by 3, the remainder is 2. What is the least possible value of x + y?

A) 7

B) 19

C) 38

D) 53

E) 89

This question can be quickly solved using an important property that can be applied to a variety of remainder questions on the GMAT. Before we examine that property, however, let me ask you an easier question:

When positive integer N is divided by 148, the remainder is 94. What is one possible value of N?

When I ask this question in class, 242 is often the first number suggested by students. If 242 was the first number you thought of, then you probably had great difficulty solving the question at the top of this article. The problem is this: while 242 does, indeed, meet the given condition (242 divided by 148 equals 1 with remainder 94), there’s a much easier number that should come to mind first. That number is 94. Notice that 94 divided by 148 equals 0 with remainder 94. Also notice that it takes no mental math to come up with this possible value of N.

A lot of GMAT remainder questions feature information about the remainder when some integer, say N, is divided by another integer. In these cases, it’s often useful to list possible values of N by applying the following rule:

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

Example: When positive integer J is divided by 7, the remainder is 2. So, the possible values of J are: 2, 2+7, 2+2(7), 2+3(7), 2+4(7), etc.

Once we evaluate these, we find that the possible values of J are: 2, 9, 16, 23, 30, etc.

Now back to the original question:

x and y are positive integers. When x is divided by 11, the remainder is 5, and when x is divided by 34, the remainder is 27. When y is divided by 17, the remainder is 11, and when y is divided by 3, the remainder is 2. What is the least possible value of x + y?

A) 7

B) 19

C) 38

D) 53

E) 89

When x is divided by 11, the remainder is 5: So, the possible values of x are: 5, 16, 27, 38, etc.

When x is divided by 34, the remainder is 27: So, the possible values of x are: 27... STOP. Since both lists include 27, the smallest possible value of x is 27.

When y is divided by 17, the remainder is 11: So, the possible values of y are: 11, 28, 45, etc.

When y is divided by 3, the remainder is 2: So, the possible values of y are: 2, 5, 8, 11...STOP. Since both lists include 11, the smallest possible value of y is 11

Since the smallest possible values of x and y are 27 and 11 respectively, the smallest possible value of x + y is 38. So, C is the correct answer to the original question.

The Big Takeaway

When solving remainder questions on the GMAT, you can sometimes save yourself a lot of work by listing possible values and, more importantly, by beginning with the smallest possible value.