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 2i.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.




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