# How to find Remainders – Basics

## Basics of Calculating Remainders

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In this post we will have a look at the basics of calculating remainders like the remainder of a sum, remainder of a product and understand what is meant by negative remainders. We will see how to use these concepts to calculate the remainders of large numbers.

### What is a Remainder?

Remainder is a number that is left over when one number, say N, is divided by another number, say D. i.e If the number N is divided by another number D, it gives Quotient Q and Remainder R. We represent this as N = Q x D + R. Where Q and R are the Quotient and Remainder respectively.

For example, 22 when divided by 9, gives a quotient 2 and remainder 4. So, 22 = 2 x 9 + 4.

Now what is the remainder of the product of 32, 64, 96 when divided by 11 . Do we have to calculate the product of all the numbers and divide it by 11 to find its remainder?

How do we find the remainder of 28 raised to 1024 (${28^{1024}}$) when divided by 9?

Lets see how to solve these kind problems,

### Remainder of Sum

Let N1, N2, N3… be numbers which when divided by a divisor D, give quotients Q1, Q2, Q3… and remainders R1, R2, R3… respectively.

Now, Remainder of sum of N1, N2, N3… when divided by D is the remainder obtained by dividing the sum of R1, R2, R3… by D.

For example,

$Remainder(\frac{28\ +\ 29\ +\ 30}{9})$ => $Remainder(\frac{1\ +\ 2\ +\ 3}{9})$ => 6

where,

1 , 2 and 3 are the remainders obtained when each of the numbers 28, 29 and 30 are divided by 9 respectively.

### Remainder of Product

Let N1, N2, N3… be numbers which when divided by a divisor D, give quotients Q1, Q2, Q3… and remainders R1, R2, R3… respectively.

Now, Remainder of product of N1, N2, N3… when divided by D is the remainder obtained by dividing the product of R1, R2, R3… by D.

For example,

$Remainder(\frac{22\ *\ 23\ *\ 24}{7})$ => $Remainder(\frac{1\ *\ 2\ *\ 3}{7})$ => 6

where,

1 , 2 and 3 are the remainders obtained when each of the numbers 22, 23 and 24 are divided by 7 respectively.

Find the remainder when 28 to the power of 1024 is divided by 9.

$Remainder(\frac{28^{1024}}{9})$ => $Remainder(\frac{28\ *\ 28\ *\ 28\ …\ (1024\ times)}{9})$

Here, 28 gives remainder 1 when divided by 9. Hence,

$Remainder(\frac{28\ *\ 28\ *\ 28\ …\ (1024\ times)}{9})$ => $Remainder(\frac{1\ *\ 1\ *\ 1\ …\ (1024\ times)}{9})$ => 1

### What is a Negative Remainder?

Consider an example,

26 when divided by 9, gives remainder 8. Hence, we have,

`26 = (9 x 2) + 8`

which is same as

`26 = (9 x 3) `

**- 1**

In the above example 8 is the positive remainder and -1 is the negative remainder of 26 when it is divided by 9.

Lets look at an example to understand how this concept is helpful in finding the remainders of large numbers.

**Problems on Negative Remainders**

1. What is the remainder when the product of 32, 64 and 96 is divided by 11.

Here,

32 = 11 x 2 + 10 or 32 = 11 x 3 – 1 => Positive remainder of 32 is 10 and its Negative remainder is -1

64 = 11 x 5 + 9 or 64 = 11 x 6 – 2 => Positive remainder of 64 is 9 and its Negative remainder is -2

96 = 11 x 8 + 8 or 96 = 11 x 9 – 3 => Positive remainder of 96 is 8 and its Negative remainder is -3

Using the negative remainders,

$Remainder(\frac{32\ *\ 64\ *\ 96}{11})$ => $Remainder(\frac{-1\ *\ -2\ *\ -3}{11})$ => $Remainder(\frac{-6}{11})$ => -6

Here, we see that the remainder is negative. The corresponding positive remainder can be obtained by adding the divisor 11 to -6, which is equal to 5.

Hence $Remainder(\frac{32\ *\ 64\ *\ 96}{11})$ = 5

2. What is remainder of 2 to the power of 96 when divided by 9?

$Remainder(\frac{2^{96}}{9})$ => $Remainder(\frac{(2^3)^{32}}{9})$ => $Remainder(\frac{8^{32}}{9})$

8 when divided by 9 gives -1 as the negative remainder. Hence,

$Remainder(\frac{8^{32}}{9})$ => $Remainder(\frac{(-1)^{32}}{9})$ => 1

Therefore, $Remainder(\frac{2^{96}}{9})$ = 1.

## Comments

## Praveen Kumar Nandigam

Hi Sujji,

Video is very impressive and keep up the hard work buddy

## basant

the helped a lot in solving problems