SAT Remainder Questions

Divisibility, Factors, and Multiples

The SAT often asks questions about remainders. These questions require you to work with divisibility, factors, and multiples. Here is a typical SAT remainder question.

If is an integer, what is the remainder when is divided by 4?

This problem is a perfect candidate for the “pick a number” strategy. Suppose is 1. Then . Now the question becomes “What is the remainder when 9 is divided by 4?”

Well, 4 divides evenly into 9 two times, with 1 left over. So, the remainder is 1.

You should double check that you get the same remainder when is, say, 2. In that case, , and 4 divides evenly into 13 three times with, again, 1 left over. So we can be confident that the correct answer is 1.

Let’s look at a more clever, potentially time-saving solution.

We know that is a multiple of 4 (because it has a factor of 4). It is therefore divisible by 4 and has a remainder of 0 (go ahead and try it with a few values of ). We can thus ignore it and concentrate solely on the remainder when 5 is divided by 4, which is 1.

Challenge Question

Test your skills with this more challenging SAT remainder question.

When 26 is divided by the positive integer , the remainder is 2. For how many different values of is this true?