Theory of numbers is wonderful. Here we see the unit digit patterns of exponents of numbers. As we go on incrementing the exponent of a number the unit digits of the results follow regular patterns. Each number has its own song. Now its time to listen to those songs:
1^{1} = 1

2^{1} = 2

3^{1} = 3

4^{1} = 4

5^{1} = 5

6^{1} = 6

1^{2} = 1

2^{2} = 4

3^{2} = 9

4^{2} = 16

5^{2} = 25

6^{2} = 36

1^{3} = 1

2^{3} = 8

3^{3} = 27

4^{3} = 64

5^{3} = 125

6^{3} = 216

2^{4} = 16

3^{4} = 81

4^{4} = 256

5^{4} = 625

6^{4} = 1296
 
2^{5} = 32

3^{5} = 243
 
2^{6} = 64

3^{6} = 729
 
2^{7} = 128

3^{7} = 2187
 
2^{8} = 256

3^{8} = 6561

7^{1} = 7

8^{1} = 8

9^{1} = 9

10^{1} = 10

7^{2} = 49

8^{2} = 64

9^{2} = 81

10^{2} = 100

7^{3} = 243

8^{3} = 512

9^{3} = 729

10^{3} = 1000

7^{4} = 1701

8^{4} = 4096

9^{4} = 6561
 
7^{5} = 11907

8^{5} = 32768
 
7^{6} = 83349

8^{6} = 262144
 
7^{7} = 583443

8^{7} = 2097152
 
7^{8} = 4084101

8^{8} = 16777216

From the above two tables we can observe the unique feature of
patterns of unit
For 1, the pattern is 1,1,1,… here frequency is 1 and the pattern is 1.Any power yields a
unit digit 1.
digits of the exponent results. (Observe the unit digits in bold
font)
The unit digits of exponentresults of each number follow a unique
pattern:
For 1, the pattern is 1,1,1,… here frequency is 1 and the pattern is 1.Any power yields a
unit digit 1.
For 2, the unit digit pattern is 2,4,8,6,2,4,8,6,… here frequency is 4 and the pattern is 2,4,8,6
For 3, the unit digit pattern is 3,9,7,1,3,9,7,1,… here frequency is 4 and the pattern is 3,9,7,1.
For 4, the unit digit pattern is 4,6,4,6… here frequency is 2 and the pattern is 4,6. Odd power results in unit digit 4 and even power results in unit digit 6.
For 9, the unit digit pattern is 9,1,9,1… here frequency is 2 and the pattern is 9,1. Odd power results in unit digit 9 and even power results in unit digit 1.
For 5, all exponents result in the unit digit 5 and similarly, for 0, all exponents result in the unit digit 0.
The following table gives the details of patterns and frequencies of numbers with unit digits from 0 to 9:
Unit digit

Pattern of unit digits of exponentresults

Frequency of patterns

1

1

1

2

2,4,8,6

4

3

3,9,7,1

4

4

4,6

2

5

5

1

6

6

1

7

7,9,3,1

4

8

8,4,2,6

4

9

9,1

2

0

0

1

Beauty here is the 'Frequencies of Patterns' also follows a pattern 1,4,4,2,1.
Now we will concentrate on our regular business of reviewing possible exam Q’s on this topic. We can apply this theory for finding unit digits of big exponents:
Now we will concentrate on our regular business of reviewing possible exam Q’s on this topic. We can apply this theory for finding unit digits of big exponents:
Find the unit digits of the following numbers:
Q1) 2^{73}
Solution: for 2, the frequency is 4. As 72 is a multiple of 4, leave up to 72 and consider 7372 which is 1 for finding the unit digit of the result. 2^{1} results in unit digit 2. Hence answer is 2.
Q2) 3^{231}
Solution: for 3, the frequency is 4. we can consider 231228 which is 3 for finding the unit digit of the result. 3^{3} results in unit digit 7. Hence answer is 7.
Q3) 19^{19}
Solution: for 9, the frequency is 2. Odd power results in unit digit 9 and even power results in unit digit 1. Here the power is 19 which is odd and hence answer is 1.
Q4) 129^{58}^{ }* 64^{29}
Consider the first number 129^{58}. As the power is even, it results in a unit digit 1.
Go to the second one 64^{29}. As the power is odd, it results in a unit digit 4.
Unit digit of the result = 1 * 4 = 4
Q5) 12^{12}^{ }+ 13^{13}^{ }+ 14^{14}^{ }+ 15^{15}
Unit digit of 12^{12}^{ }: As 12 is multiple of 4, the unit digit is pattern end number ie.,6
Unit digit of 13^{13}^{ }is 3
Unit digit of 14^{14}^{ }is 6
Unit digit of 15^{15}^{ }is 5
Unit digit of the result is that of 6+3+6+5 ie., 0
Link for the next post in the Magic Series is here:
http://mathbyvemuri.blogspot.in/2012/12/magicofnumbersii.html
Link for the next post in the Magic Series is here:
http://mathbyvemuri.blogspot.in/2012/12/magicofnumbersii.html
Q3) shouldnt it be 9. Odd exponents results in units digit 9
ReplyDeleteTq btw. I find this useful and interesting for GMAT.
ReplyDelete