SAT Remainder Questions: Divisibility, Factors, and Multiples

SAT Math Tutorial

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.

Sample Problem

Question source

If n is an integer, what is the remainder when 4n+5 is divided by 4?







This problem is a perfect candidate for the “pick a number” strategy. Suppose n is 1. Then 4n+5=4(1)+5=9. 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 n is, say, 2. In that case, 4n+5=4(2)+5=13, 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 4n 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 n). 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 k, the remainder is 2. For how many different values of k is this true?

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