Remainder

So far we have been introduced to remainders. In mathematics section of any exam, questions on remainder are quite common. Therefore, you should know this topic in and out. So, let’s get started.

Various Ways To Calculate Remainders:

1. Method of finding Remainder by cyclicity.
2. Method of finding Remainder by negative remainder concept
3. Method of finding Remainder by Euler’s, Fermat’s little theorem , Wilson’s theorem
4. Method of finding Remainder: “Remainder theorem” ( in case of polynomials )
5. Method of finding Remainder: Chinese theorem
6. Some important formats to find remainders

In this article, we’ll study the first two ways to find remainders of a given number.

1. Method to find Remainder by Cyclicity:

Remainders repeat themselves in the same pattern after a certain length. This helps in solving questions quickly.

Let us understand it with the help of an example:

Example: Find the remainder of

Solution:

Step 1:

Check what pattern is made by remainders of this particular set of divisor and dividend.

Step 2:

Now write the power in the form of  3n+ k

10 = 3*3+1

Solution:

 Step 1 :

Check what pattern is made by remainders of this particular set of divisor and dividend .

Now we know the cycle is of 6

Step 2 :

Now write the power in the form of  6n+ k

500 = 6*83+2

Given below are few tips based on the above method,

Tooltips:

1. To find remainder with the help of Negative Remainder concept:

Concept of negative remainder is very useful to reduce calculation. Let us understand the negative remainder and its use.

If we have to divide 120 by 11 then remainder obtained is 10

Dividend = divisor * quotient + remainder

N = D*Q+R

120 = 11*10+10

Or in other way we can write

120 = 11*11 – 1 hence remainder is (–1).

So R can be either 10 or  (-1)

So we can say that when a number N is divided by D then remainder obtained it R which can also be written as R’= (R -D )

Let us see the use of negative remainder concept:

Example: Find the remainder of

Solution:

Method 1:

R = 10

R’ = R-D=10-15= (-5)

443  = 15*29+8

R = 8

R’ = R-D=8-15= (-7)

{This method looks a little lengthy, but with a little practice it can be mastered and used everywhere}

Example: Find the remainder of Solution:

By using negative remainder concept:

EXERCISE:

Question 1. Find the remainder of

(1) 5

(2) 7

(3) 8

(4) 4

Question 2.  Find the remainder of .

(1) 5

(2) 7

(3) 8

(4) 4

Question 3. Find the remainder when 79⁶⁴⁴ is divided by 9.

(1) 5

(2) 7

(3) 1

(4) 4

Question 4. Find the remainder when 47¹²³ is divided by 7.

(1) 4

(2) 6

(3) 2

(4) 3

Question 5. Find the remainder when 2⁵⁷ is divided by 3?

a) 0

b) 1

c) 2

d) None of these