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

2 comments:

  1. Q3) shouldnt it be 9. Odd exponents results in units digit 9

    ReplyDelete
  2. Tq btw. I find this useful and interesting for GMAT.

    ReplyDelete