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