Remainders (Part-4)

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:
Answer – c
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:
Answer-b
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 :
Answer – b
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 :
Answer- a
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.