Application of Concept of Cyclicity

How to calculate unit digit if a number contains power of a power

Example 1:What will be the last digit of ${{12}^{{{23}^{45}}}}$

Solution:
To find the last digit of this type of number we will start the question from the base the base is given to be 12. It means we will see the cyclicity of 2 because the last digit depends upon the unit digit of 12 i.e. 2. We observe unit digit while calculating powers of 2
2¹ = 2
2² = 4
2³ = 8
2⁴ = 6
2⁵ = 2

Step 1: Now we know that cyclicity of last digit of 12 i.e. 2 is of 4, hence we divide the power of 12 i.e. 23⁴⁵ with 4.

Step 2: Now lets calculate the remainder of 23⁴⁵ when divided by 4 and then we will determine the last digit.

Step 3:  The remainder will be 3 because we can write remainder of 23 by 4 as 3 or -1.Hence we can write 23⁴⁵ as (-1) ⁴⁵ but odd power of -1 will be again -1 and thus 23⁴⁵ when divided by 4 will give us remainder as -1 or 3.

Hence unit digit of ${{12}^{{{23}^{45}}}}$ will be same as unit digit of 2³i.e. 8

Example 2: Find the unit digit of ${{32}^{{{25}^{95}}}}$  ?  

Solution: To find the last digit of this type of number we will start the question from the base the base is given to be 32. It means we will see the cyclicity of 2 because the last digit depends upon the unit digit of 32 i.e. 2. We observe unit digit while calculating powers of 2
2¹ = 2
2² = 4
2³ = 8
2⁴ = 6
2⁵ = 2
Step 1: Now we know that cyclicity of last digit of 32 i.e. 2 is of 4, hence the divide the power of 12 i.e. 25⁹⁵ with 4.
Step 2: Now lets calculate the remainder of 25⁹⁵ when divided by 4 and then we will determine the last digit.
Step 3:  The remainder will be 1 because we can write remainder of 25 by 4 as 1. Hence we can write 25⁹⁵ as (1) ⁹⁵ but any power of 1 will be again 1 and thus 25⁹⁵ when divided by 4 will give us remainder as 1 .
Hence unit digit of ${{32}^{{{25}^{95}}}}$   will be same as unit digit of 2¹ i.e. 2

Assignment:
Questions:
1: Find the unit digit of 4⁶⁸⁶
a) 4
b)  6
c) 2
d)  3
Answer: b
Solution: We know
4¹ = 4
4² = 6
4³ = 4
4⁴ = 4
Here, we see that powers of four repeat after a cycle of 2 i.e.
Odd power of 4 gives unit digit as 4 and even power of 4 gives unit digit as 6
So unit digit of 4⁶⁸⁶ is 6.

2 : What will be the unit digit of 65¹²³⁴
a) 5
b)  0
b)  2
d) 3
Answer: a
Solution We know
5¹ = 5
5² = 25
5³ = 125
5⁴ = 625
So, we observe that unit digit of any power of 5 is 5.Hence unit digit of 65¹²³⁴ is 5.

3:  Find the unit digit of 36⁵¹²
a) 4
b) 6
c) 2
d) 3
Answer: b
Solution We know
6¹ = 6
6² = 36
6³ = 216
So, we observe that unit digit of any power of 6 is 6.Hence unit digit of 36⁵¹² is 6.

4: Find the unit digit of 18⁵⁶⁹³
a) 4
b) 6
c) 2
d) 8
Answer: d
Solution: We observe unit digit while calculating powers of 8
Unit digit in 8¹ = 8
Unit digit in 8² = 4
Unit digit in 8³ = 2
Unit digit in 8⁴ = 6
Unit digit in 8⁵ = 8
So on dividing 5693 with 4, 1 will be the remainder and hence the last digit would be 8.