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Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Division problems often result in remainders. In this lesson, we will define the remainder and use long division of numbers and synthetic division of polynomials to compute remainders.

What Is a Remainder?

Imagine you are entertaining a group of 14 people. You have 50 muffin appetizers, and you want to serve each individual the same number of muffins. You line up 14 plates and begin to distribute the muffins, one at a time. After filling each of the 14 plates with 3 muffins a piece, you realize that you don't have enough muffins left for another round, so the 8 muffins are extras, or remainders.

The scenario just described is a division problem. We divided 50 by 14 and found that 14 goes into 50 exactly 3 times with 8 left over: 50 /14 = 3 R8. In this problem, the number 50 is the dividend, or the figure we want to divide up; 14 is the divisor, or the number we are dividing by. The quotient is how many times the divisor fits into the dividend, in this case, 3. Lastly, the remainder is what's left over, or 8.

The remainder of a division problem is an interesting part of the problem. To find the remainder of a division problem, we can use long division of numbers.

Remainders and Long Division

Most of us are used to just grabbing our calculators to perform division of numbers. However, this doesn't normally show us our remainder. If there is no calculator available, it's also useful to know how to perform division by hand or in our heads.

For instance, imagine you are at a store and come across an item you need to stock up on. You have $137 in cash, and the item is priced at $5 each. You need to know how many items you can purchase and how much money you'll have left over. One method for performing this calculation is known as long division, through which we can divide 137 by 5, or 137 / 5.

We will look at the steps involved with long division, and we will use our example of 137 / 5 to illustrate each of the steps.

Step 1: Divide your divisor into the first number of your dividend. If it doesn't fit, divide it into the first two numbers of your dividend, and so on until it fits. Place this number above your dividend. In our example, we see that 5 does not go into 1, so we divide 5 into the first two numbers, 13. We have that 5 goes into 13 two times, so we write a 2 above our dividend.

Step 2: Multiply your divisor by the number you just placed above your dividend. Place this number below your dividend, lining it up to the left and subtract. Bring down any remaining numbers. In our example, we multiply 5 by 2 to get 10. We then place 10 under our dividend lined up to the left, then subtract to get 3. There is a 7 left over in the dividend, so we bring that down.

Step 3: Repeat steps 1 and 2 using the number you just created in the bottom row as your dividend, until you don't have anymore numbers to bring down. In our example, we use 37 as our dividend and repeat steps 1 and 2. Thus, we divide 5 into 37 to get 7. We place 7 above our dividend next to the last number we place above the dividend. We then multiply our divisor, 5, by 7 to get 35. We place 35 under 37 and subtract to get 2. We have no more numbers to bring down, so 2 is our remainder.

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We see that when we divide 137 by 5, we get a quotient of 27 and a remainder of 2. The results tell us we can buy 27 of the items at $5 a piece and still have $2 left over. The $2 is our remainder.

Another Example

Let's look at one more example. Assume you and 19 of your friends win a contest. The prize money is $1,454. After you give yourself and each of your friends an equal amount of the winnings, how much is left over? In this scenario, our divisor is 20 since you are splitting the money between you and your 19 friends, so there are 20 of you. Our dividend is the amount of the prize money, $1,454. So we want to find the remainder when we divide 1,454 by 20. We will use long division to find our remainder. We set up our problem with 1,454 as our dividend and 20 as our divisor and then follow the same steps.

Twenty does not go into 1, so we need to include the next number, 4. Because 20 does not go into 14, we have to include the next number, 5. We know that 20 can go into 145 seven times, so we place 7 above the 5; 7 x 20 = 140, so we place that under the 145 and subtract. We are left with 5 and then we bring down the next number, which is 4. 20 goes into 54 two times, so we put the 2 next to the 7 and multiply 2 by 20. That gives us 40, which we subtract from 54. That leaves us with 14. Because we have no other numbers to bring down, we are done. We see that when we divide 1,454 by 20 we get a quotient of 72 with a remainder of 14. This tells us that after each of the 20 people get $72 each of the prize money, there is $14 left over.

Lesson Summary

Sometimes we want to know what we have left after dividing something up evenly among a group. A remainder in mathematics is what's left over in a division problem. In the division process, the number we want to divide up is known as the dividend, while the number we are dividing by is referred to as the divisor; the result is the quotient. We can find the remainder of a division problem by using long division. This process allows us to answer many questions about a problem involving division and is an extremely useful tool.

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