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.

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