Remainder

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What is Remainder?

The remainder, as the name says itself, is something that is left over after performing a task.

Suppose, you have 9 toffees and you have to distribute them equally among 4 of your friends.

You will begin distributing the toffees in the following way.


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Here, you can see that there is one toffee left after distribution.

This one toffee that is left cannot be further shared among the 4 of your friends.

Hence, 1 is called the ‘ remainder’ here.


Remainder in Maths

The remainder in Maths is defined as the value that is left after performing division. If a number (dividend) is not completely divided by another number (divisor), then we are left with a value once the division is performed. This value is known as the remainder.

For example, 7 cannot be completely divided by 2. Because, the nearest value we can get is 2 × 3 = 6, which is 1 less than 7.

Hence, the answer of 7 ÷ 2  is 3 with a remainder of 1. It implies that 7 can be divided by 2 into 3 parts but with 1 leftover, and is usually represented as “ 3 R 1”, where 3 is the quotient and 1 is the remainder.


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Representing Remainder

Let us divide 10 by 3 using the long division and see what the remainder and quotient are left.


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Here, the remainder is 1 and quotient is 3.

Remainders in Maths can be represented in two ways.

One way is to write the quotient and remainder with an “ R” in between them.


Dividend ÷ Divisor = Quotient R Remainder


For example, 10 ÷ 3 can be expressed as:

10 ÷ 3 = 3 R 1

Another way to represent the remainder is by expressing it as a part of the mixed fraction.

\[\frac{\text{Dividend}}{\text{Divisor }} = \text{Quotient} \frac{\text{Remainder}}{\text{Divisor}}\]

For example, 10 ÷ 3 can be written as:

\[10 \div 3 = 3 \frac{1}{3}\]


Remainder Formula

As we know,

Dividend = Divisor × Quotient + Remainder

Accordingly, the remainder formula is given as:


Remainder =  Dividend - (Divisor × Quotient)


In the above remainder formula,

  • The dividend is the number or value that is being divided.

  • The divisor is the number that divides another number.

  • The quotient is the result that is obtained after the division of two numbers.

  • The remainder is the value that is left after the division.

How to Find The Remainder?

As you must have understood what is remainder, let us now discuss how to find the remainder?

To find the remainder of a number x upon division by a divisor d, we will first find the largest multiple of divisor d that goes into, and the remainder r is the value that is left over.

x = kd + r with r < d

For Example,

Find the value of the remainder when 37 is divided by 6?

To find the value of the remainder, when 37 is divided by 6, we will look at  the multiples of 6 given below:

6 × 1 = 6

6 × 2 = 12

6 × 3 = 18

6 × 4 = 24

6 × 5 = 30

6 × 6 = 36

6 × 7 = 42

As 6 × 6 = 36 and 6 × 7 = 42, only six 6’s can go into 37. Then the value left over is 37 - 36 = 1. Hence, the value of the remainder when 37 is divided by 6 is 1.


Remainder Examples

Here are a few examples of remainders that will help you understand the term remainder in Maths in a better way.

1. A teacher had 315 chocolates. She divides all chocolate evenly among 30 students. Find

  • How many chocolates did the teacher give to each of the students?

  • How many chocolates are left with the teacher after distributing them among the student?

Solutions:

Total number of students in class = 30

Total number of chocolates teacher has = 315

Now, we will divide 315 by 30


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i) The quotient value of the above division represents the number of chocolate each student gets is 10.

Therefore,

Chocolates each students gets  = 10

ii) The remainder value of the above division represents the number of chocolate left with the teacher.

Therefore,

Chocolates left with the teacher after distribution = 15


2. What is the remainder when 53 is divided by 8?

Solution: To find the value of the remainder, when 53 is divided by 8, we consider the multiples of 8.

8 × 1 = 8

8 × 2 = 16

8 × 3 = 24

8 × 4 = 32

8 × 5 = 40

8 × 6 = 48

8 × 7 = 56

As 8 × 6 = 48 and 8 × 7 = 56, only six 8’s can go into 53. Then the value left over is 53 - 48 = 5. Hence, the value of the remainder when 53 is divided by 8 is 5.

FAQ (Frequently Asked Questions)

Q1. What are the Different Properties of the Remainder in Maths?

Answer: The different properties of the remainder in Maths are as follows:

  • The value of the remainder is always greater than the divisor. If the value of the remainder is less than or equal to the divisor, it implies that division is incomplete.

  • The value of the remainder can be either lesser, equals to, or greater than the value of the quotient.

  • If a number (divisor) completely divides another number ( dividend), the value of the remainder is 0.

Q2. Can the Number 0 can be a Remainder?

Answer: Yes, the number 0 can be a remainder when the divisor divides the dividend completely. For example, the value of the remainder when 30 is divided by 6 is 0.

Q3. What is the Remainder of 19 Divided by 6?

Answer: 19 divided 6 is 3 R 1

Where 3 is the quotient and 1 is the remainder.

Hence, the remainder of 19 divided by 6 is 1.

Q4. What is an Example of the Remainder?

Answer: When 29 chocolates are distributed equally among 9 children, each child gets 3 chocolates and 2 chocolates are left undistributed. Here, 2 is the remainder.