SAT cookie remainder
Can We Solve Remainder Problems

Without Using Long Division?

For the next three days I would like to help you to understand math problems involving remainders. These types of problems appear on standardized tests such as the ACT, GRE, and SAT math subject tests.

ACT and GRE math problems with remainders seem to give students a difficult time. This is mostly because a fundamental step in solving the problem is often missed – performing long division.

You cannot solve a remainder problem by simply dividing in your calculator. There are, however, calculator algorithms that can give you the answer very quickly. We will talk about these a bit later in this post.

A common error that students make is to perform a division calculation on their calculator, and simply take the first number after the decimal point and use this digit as the answer to the problem. This will usually result in a wrong answer.

Example

Suppose we are asked “Find the remainder when 14 is divided by 4.”

In your calculator, 14/4 = 3.5.

Penciling the number 5 as your answer would be incorrect – seriously incorrect. This “5” is actually part of the answer to the question “What is 14 divided by 4?” But it has nothing to do with the remainder.

Students that are currently scoring in the mid-range in ACT/GRE math seem to have the biggest problem with remainder questions. Many cannot perform long division correctly, and some do not even realize that long division is needed. This keeps many students from getting their scores to the next level.

Let’s take a look at how we can solve the problem of finding the remainder when 14 is divided by 4 in three different ways. This problem may seem very basic for some of you, but let’s go over it anyway to make sure we have a strong foundation before solving more difficult remainder problems.

Method 1 – Long Division: So 4 goes into 14 three times with 2 left over. In other words, 14 = 4(3) + 2. So we see that the remainder when we divide 14 by 4 is 2.

Let’s break this down step by step:

SAT math long division explanation

Count the number of groups of 4 objects that can be formed from 14 objects. Place this number over the 4 that is inside the division symbol.

SAT math long division explanation

Multiply this resulting  number 3 by the divisor 4. Place this number below the 4 that is inside the division symbol.

SAT math long division explanation 2

Subtract the resulting number 12 from the dividend 14.

SAT math long division explanation 3

The quotient 3 appears above the division symbol and the remainder 2 appears at the bottom.

Below is a visual representation of 14 divided by 4. In the figure below we are grouping 14 objects 4 at a time. Note that we wind up with 3 groups (the quotient) and 2 extra objects (the remainder).

sat math long division remainder diagram

Method 2 – First Calculator Algorithm:

Step 1: Perform the division in your calculator:

14/4 = 3.5

Step 2: Multiply the integer part of this answer by the divisor:

4*3 = 12

Step 3: Subtract the above result from the dividend to get the remainder:

14 – 12 = 2.

Method 3 – Second Calculator Algorithm:

Step 1: Perform the division in your calculator:

14/4 = 3.5

Step 2: Subtract off the integer part of this result:

ANS – 3 = .5

Step 3: Multiply this result by the divisor:

4*ANS = 2.

Note that these calculator algorithms work exactly the same no matter how large the numbers are that you are dividing.

Exercise

As an additional exercise you should try to find the remainder when 15,216 is divided by 73 by using one of these calculator algorithms. Figure out which of the two algorithms you prefer. You can find solutions here: SAT Remainder Problems – Part 2

More Remainder Problems with Explanations

If you are preparing for the ACT, the GRE, or an SAT math subject test, you may want to take a look at one of the following books.

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