UNIT DIGIT

 UNIT DIGIT CONCEPT:

Unit Digit Concept for Number 2:

2^1=2

2^2=4

2^3=8

2^4=16

2^5=32

2^6=64

From the above Concept of unit digit for 2, we can understand the Unit Digit of powers  2 starts to repeat after the 4th step. Number 2 has  4 unique powers and after which they repeat. Similarly, each number has unique digits up to a certain power and after which the powers tend to repeat.

Power Cycle of No 3: 4 cycles;3^1=3,3^2=9,3^3=27,3^4=81,3^5=243

Power Cycle of No 4: 2 cycles;4^1=4,4^2=16,4^3=64

Power Cycle of No 5: 1 cycle;5^1=5,5^2=25

Power Cycle of No 6: 1 cycle;6^1=6,6^2=36

Power Cycle of No 7: 4 cycles;7^1=7,7^2=49,7^3=343,7^4=2401,7^5=16807

Power Cycle of No 8: 4 cycles;8^1=8,8^2=64,8^3=512,8^4=4096,8^5=32768

Power Cycle of No 9: 2 cycles;9^1=9,9^2=81,9^3=729

Example 1:

Find the unit digit of 14^123

Solution:

The last or Unit digit of the given problem is 4, as we know 4 has 2 cycles, which means we have only 2 unique values for its powers after which the values start to repeat.

So Divide 123 by 2 so we  can actually understand if we would get the first or the second power 

123/2, We get the quotient as 61 and the remainder as 1, If the remainder is '1' we need to understand the first step is the unit digit of the given value.

so our answer for 14^123 is the same answer as 4^1=4.

for instance, if we had to find the value of 14^124

we would have divided 124/2 to get the quotient as 62 and the remainder as 0 . This stands for the 2nd step getting repeated because 124 is divisible by 2 and since 4 has 2  unique unit digits we had to divide it by 2. So we get our unit digit as the 6 since 124 leads to the repetition of the second power cycle.

Example 2:

Find the Unit digit of 33^422

The unit digit in the given value is 3 and 3 has 4 unique powers.

so we divide by 4

422/4, Quotient =105 ,Remainder =2 

So second power repeats 

3^2=9,

so 33^422 has the unit digit as 9.

Example 3:

Find the unit digit of 24*33*46*29

To find the unit digit of such numbers without powers, just multiply the unit digits of the given numbers

4*3*6*9= 8

so 24*33*46*29 has the unit digit as 8.