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Remainders: Part-4
In this article, for calculating remainders we will use concept of cyclicity.
Let’s take an example of questions based on concept of remainders.

Example
Find the remainder when 47123 is divided by 7?
Solution:
We will do this question step by step
Step 1: We divide 47 with 7 to get remainder as 5.
Step 2: Here in the question 47 is multiplied by itself 123 times. Now understand every time when we will divide 47 with 7 the remainder will be 5.
Step 3: Hence we can say finding remainder of 47123is equivalent to finding remainder of 5123 when divided by 7.
So we start observing powers of 5.
52 gives remainder 4.
53 gives remainder 6.
54 gives remainder 2.
55 gives remainder 3.
56 gives remainder 1.
Hence after 6th power remainders will start repeating. So we have a cycle of 6 and thus we divide 123 by 6

i.e. 123= 6(20) + 3
Hence 5 123

-> 5 6(20) +3

-> 5 6(20) .5 3 -> 1.6  =6   (As 56 gives remainder 1 and 53 gives remainder 6)
Hence the remainder of 47123 when divided by 7 is 6.

Example
Find the remainder when 79644 is divided by 9 ?
Solution:
We will do this question step by step
Step 1: We divide 79 with 9 to get remainder as 7.
Step 2: Here in the question 79 is multiplied by itself 644 times. Now understand every time when we will divide 79 with 9 the remainder will be 7.
Step 3: Hence we can say finding remainder of 79644 is equivalent to finding remainder of 7644 when divided by 9.
So we start observing powers of 7.
72 gives remainder 4.
73 gives remainder 1.
Hence after 3rd power remainders will start repeating. So we have a cycle of 3 and thus we divide 644 by 3 i.e. 644= 3(214) + 2
Hence 7 644  -> 7 3(214) + 2 -> 73(214) .72-> 1.4  = 4   (As 73 gives remainder 1 and 72 gives remainder 4)
Hence the remainder of 79644 when divided by 9 is 4.

Assignment:
QUESTIONS

1. Find the remainder when 257 is divided by 3?
a)0
b)1
c)2
d) None of these
Solution:
So we start observing powers of 2.
21 gives remainder 2.
22 gives remainder 1.
23 gives remainder 2.
24 gives remainder 1.
Hence we can observe odd power of 2 gives remainder 2 and even power gives remainder 1.
Hence 257 will give remainder 2 when divided by 3.

2. Find the remainder when 391 is divided by 5 ?
a)1
b)2
c)3
d)4
Solution:
So we start observing powers of 3.
31 gives remainder 3.
32 gives remainder 4.
33 gives remainder 2.
34 gives remainder 1.
Hence after 4th power remainders will start repeating. So we have a cycle of 4 and thus we divide 91 by 4 and 391 will give same remainder as 33 i.e. 2.
Hence the remainder of 391 when divided by 5 is 2.

3. What is the remainder when 1548 is divided by 4?
a)1
b)2
c)3
d)0
Solution :
Finding remainder of 1548 is equivalent to finding remainder of 348 when divided by 4.
So we start observing powers of 3.
31 gives remainder 3.
32 gives remainder 1.
33 gives remainder 3.
34 gives remainder 1.
Hence we can observe odd power of 3 gives remainder 3 and even power gives remainder 1.
Hence 348 will give remainder 1 when divided by 4.
Hence the remainder of 348 when divided by 4 is 1.

4. Find the remainder when 1348is divided by 9?
a)1
b)2
c)3
d)4
Solution :
Finding remainder of 1348 is equivalent to finding remainder of 448 when divided by 9.
So we start observing powers of 4.
41 gives remainder 4.
42 gives remainder 7.
43 gives remainder 1.
Hence after 3rd power remainders will start repeating. So we have a cycle of 3 and we know 48 is divisible by 3 so answer will same as remainder of 43.
Hence the remainder of 1348 when divided by 9 is 1.

5. What is the remainder when 2871 is divided by 7?
a)1
b)2
c)3
d)0
Solution :