## Sunday, 9 October 2011

### Magic of Numbers-1 (Unit digit Patterns of number exponents)

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:

 11 = 1 21 = 2 31 = 3 41 = 4 51 = 5 61 = 6 12 = 1 22 = 4 32 = 9 42 = 16 52 = 25 62 = 36 13 = 1 23 = 8 33 = 27 43 = 64 53 = 125 63 = 216 24 = 16 34 = 81 44 = 256 54 = 625 64 = 1296 25 = 32 35 = 243 26 = 64 36 = 729 27 = 128 37 = 2187 28 = 256 38 = 6561

 71 = 7 81 = 8 91 = 9 101 = 10 72 = 49 82 = 64 92 = 81 102 = 100 73 = 243 83 = 512 93 = 729 103 = 1000 74 = 1701 84 = 4096 94 = 6561 75 = 11907 85 = 32768 76 = 83349 86 = 262144 77 = 583443 87 = 2097152 78 = 4084101 88 = 16777216

From the above two tables we can observe the unique feature of patterns of unit
digits of the exponent results. (Observe the unit digits in bold font)
The unit digits of exponent-results 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 exponent-results 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:

Find the unit digits of the following numbers:

Q1) 273
Solution: for 2, the frequency is 4. As 72 is a multiple of 4, leave up to 72 and consider 73-72 which is 1 for finding the unit digit of the result. 21 results in unit digit 2. Hence answer is 2.

Q2) 3231
Solution: for 3, the frequency is 4. we can consider 231-228 which is 3 for finding the unit digit of the result. 33 results in unit digit 7. Hence answer is 7.

Q3) 1919
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) 12958 * 6429
Consider the first number 12958. As the power is even, it results in a unit digit 1.
Go to the second one 6429. As the power is odd, it results in a unit digit 4.
Unit digit of the result = 1 * 4 = 4

Q5) 1212 + 1313 + 1414 + 1515
Unit digit of 1212 : As 12 is multiple of 4, the unit digit is pattern end number ie.,6
Unit digit of 1313 is 3
Unit digit of 1414 is 6
Unit digit of 1515 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/magic-of-numbers-ii.html