Numbers

CYCLICITY TABLE

1	1
2	4
3	4
4	2
5	1
6	1
7	4
8	4
9	2
10	1

Cyclicity of a number represents total number of different numbers at the unit place

e.g

cyclicity of 3 is 4 

meaning 3^n will end 4 different value at the unit place for different values of n

3^1-3 

3^2-9

3^3-27 unit digit is 7

3^4-81 unit digit is 1

3^5-243 unit digit is 3

3^6-729 unit digit is 9

so it gives us a pattern of 3,9,7,1 (4 different numbers)

Pattern for all numbers

1-1

2-2,4,8,6

3-3,9,7,1

4-4,6

5-5

6-6

7-7,9,3,1

8-8,4,2,6

9-9,1

Finding unit digit

e.g 

Find the unit digit of 3^514

step1- divide 514 by 4 (because 3 has cyclicity of 4) and find the reaminder. i.e. 2

step2- so 2nd number in pattern of 3 is 9

Ans-9

Try other examples

1)find the last digit of 457^177.
2)find the last digit of 19^146.
3)find the last digit of 904^507.

Remember we have to focus only on last digit.so unit digit of 904^507 is same as 4^507.

Finding last two digit

1)for odd numbers(1,3,7,9)

Rule 1- ending with 1.

341^129

second last digit is - second digit of base (4) * last digit of power(9)

4*9=32 so second last digit will be 2

so last two digit will be 21.

Note- with help of pattern table we can convert all odd number into 1 as a last digit using power

Rule 2-

find last two digit of 1739^768

39^768

or (39^2)^384 which is xx21^384 , now we can follow Rule 1.

so Ans will be 81

2)For even numbers (2,4,6,8)

Rule 1- Any number ending with 2 power 5 ends with 32.

Rule 2- Any number ending with 2 power 10 ends with 24

Rule 3- xxxx24^even ends with 76.

Rule 4- xxxx24^odd ends with 24.

Rule 5- xxxx76^n ends with 76.

e.g.

find last two digit of 1372^482

72^482

72^480 * 72^2

(72^10)^48 * 72^2

xx24^48(By Rule-2) * xx84

76(By Rule-3) *84

6384 so last two digits will be 84.