# Remainders – Possible values

Here’s an example of a quantitative reasoning question you might encounter on the GRE. Try to answer it before continuing.

When positive integer R is divided by 7, the remainder is 1. When R is divided by 76, the remainder is 43. What is the smallest possible value of R?

How did you do?

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

Q is a positive integer. When Q is divided by 10, the remainder is 3. What are three possible values of Q?

If you’re like most students, the first value of Q you thought of was 13, then perhaps 23, and then 33.

What’s missing here? Does your list include the number 3?

Many students overlook 3 as a possible value of Q, even though it meets the given criteria: 3 divided by 10 equals 0 with remainder 3.

On the GRE, you may be given information regarding the remainder when some integer is divided by another integer, and in many cases, a quick way to find the solution is to begin listing possible values.

When it comes to listing possible values, the rule is as follows:

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 8, the remainder is 1. So, the possible values of W are: 1, 1+8, 1+2(8), 1+3(8), 1+4(8), 1+5(8). . .

When we simplify, we can see that the possible values of W are: 1, 9, 17, 25, 33, 41 and so on

The main point I want to emphasize here is that you can sometimes save yourself a lot of work by remembering to consider the smallest possible value (in the above example, the smallest possible value of W is 1).

When positive integer R is divided by 7, the remainder is 1. When R is divided by 76, the remainder is 43. What is the smallest possible value of R?

For each piece of given information, we’ll list the possible values of R.

a) When R is divided by 7, the remainder is 1.

Some possible values of R: 1, 8, 15, 22, 29, 36, 43, 50, 57,. . .

b) When R is divided by 76, the remainder is 43.

Some possible values of R: 43, . . .

At this point, we can stop, since both lists include 43. So, we can plainly see that the smallest possible value of R is 43.

Please keep in mind that there are other instances in which it’s useful to list possible values (when given a remainder). In those instances, be sure that your list includes the smallest possible value, also known as the remainder. Doing so, may save you some time.