It is very difficult to find the last digit of a number by usual traditional method. In number theory finding the last digit or finding the remainder when any number is divided by 10 is having more importance and also one of interesting problem.

However if u remember and used the following rule, we can easily find the last digit.

Last digit of the number of the form m

^{n}

m n=1 n=2 n=3 n=4 n=5

0 0 0 0 0 0

1 1 1 1 1 1

2 2 4 8 6 2

3 3 9 7 1 3

4 4 6 4 6 4

5 5 5 5 5 5

6 6 6 6 6 6

7 7 9 3 1 7

8 8 4 2 6 8

9 9 1 9 1 9

Last digit of m

^{n}is changing upto n=4 when n=5, the last digit is repeated. Hence the last digit is periodic with 4

METHOD: To find the last digit of number m

^{n}Divide n by 4, If the remainder is r, then the last digit of the number is (the last digit of given number)^r. otherwise if remainder is zero, then the last digit of number is last digit of m^4 i.e (last digit of given number)^4.

ILLUSTRATIONS:

1) The last digit of the number

N=(2003)

^{5003}+(6007)

^{905}–(209)

^{608}=3

^{3}+7

^{1}-9

^{4 }=14-1=13, the last digit is 3.

2)Find the remainder when

(608)^{1829} is divided by 10

Sol: Finding the remainder when any number is divided by 10 is same as finding the last digit of the number , since when any number is divisible by 10 the remainder is the last digit of the given number. Last digit of 608 is 8 and when 1829 is divided by 4 , remainder is 1

^{1}(by rule )

3)Find the last digit of 4^1328

Here when 1328 is divided by 4 , Remainder is 0 , so the last digit of 4^1328 is last digit of 4^4 i.e 6

FINDING THE LAST DIGIT ANY NUMBER OF THE FORM m^n ENDING WITH 0,1,2,3,4,5,6,7,8,9

a)The last digit of 5^1 is 5

5^2 is 5

5^3 is 5

Hence the last digit of 5^any number is 5

b)The last digit 6^1 is 6

6^2 is 6

6^3 is 6

and so on.

Hence the last digit of 6^any number is 6

c)The last digit of

11^1 is 1

11^2 is 1

11^3 is 1

11^4 is 1 and so on

The last digit of (any number ending with 1)^any number is 1

d)Similarly we can show that the last digit of (any number any number ending with 0)^any number is 0

e)Last digit of 2^1 is 2

2^2 is 4

2^3 is 8

2^4 is 16 ie 6

2^5 is 32 ie 2

2^6 is 64 i.e 4

2^7 is 128 i.e 8

2^8 is 256 i.e 6

In general we can see that the last digit of (any number ending with 2)^any number is 2,4,8,6

when power n is odd , the last digit is 2 and 8; when the power is even, the last digit is 4 and 6

**2**

^{2n}**≡**

**6(mod 10) when n is even**

**≡** **4(mod 10) when n is odd**

**f)****3 ^{2n} ≡ 1(mod 10) when n is even .**

**≡****9(mod 10) when n is odd**

g) We can also see that last digit of (any number ending with 7)^any number is 1 when n is odd and 9 when n is even

**7**

^{2n}≡1(mod 10) when n is even. **≡** **9(mod 10) when n is odd**

**h)The last digit of (any number ending with 8)^any number is is 8, 6 or 4**

** ****8 ^{2n} ≡ 6(mod 10) when n is even. **

**≡ ****4(mod 10) when n is odd.**

**j)****9 ^{n} ≡ 1(mod 10) when n is even **

**≡** **9(mod 10) when n is odd**

**To find the unit or last digit remember the following:(Try to prove)**

**a)5 ^{n}≡5(mod 10) b)6^{n} =6(mod 10)**

**c)4 ^{n} ≡ 6(mod 10) when n is even .**

**≡ ** **4(mod 10) when n is odd **

**d)9 ^{n} ≡ 1(mod 10) when n is even **

**≡ ** **9(mod 10) when n is odd**

**e)3 ^{2n} ≡ 1(mod 10) when n is even .**

**≡ ****9(mod 10) when n is odd**

**f)2 ^{2n} ≡ 6(mod 10) when n is even **

**≡ ****4(mod 10) when n is odd**

**g)8 ^{2n} ≡ 6(mod 10) when n is even. **

**≡ ****4(mod 10) when n is odd.**

**h) 7 ^{2n} ≡1(mod 10) when n is even.**

**≡ ****9(mod 10) when n is odd i.e **

**When n is positive integer ,**

the last digit of 5^{n} is 5 ; the last digit in 6^{n} is 6.

The last digit in 3^{4n} is 1; The last digit in 9^{n} is 1 or 9 according as n is odd or even. The last digit of 11^{n} is 1, 6^{n} is 6 , 4^{n} is 4 or 6 according as n is odd or even.

**examples: **

Unit digit of 17^{189} =unit digit of 7^{189} =7

Unit digit of 28^{200} =unit digit of 8^{200} =6

Unit digit of 43^{20} =unit digit of 3^{20} =1

Unit digit of 79^{27} =unit digit of 9^{27} =9

Unit digit of 84^{37} =unit digit of 4^{37} =4

^{}

## 11 comments:

I want not agree on it. I assume nice post. Particularly the title attracted me to be familiar with the unscathed story.

Good fill someone in on and this post helped me alot in my college assignement. Thank you on your information.

Good fill someone in on and this fill someone in on helped me alot in my college assignement. Thank you seeking your information.

Thanks a lot for the post. :) Nice n helpful.

Thanks a lot man, it was very educational. It cleared up all my doubts. Thanks again!!

This stuff really works

Aw, this was a really quality post. In theory I'd like to write like this too - taking time and real effort to make a good article... but what can I say... I procrastinate alot and never seem to get something done.

Thanks!

Thank u, Good explanation and easy to understand.

Good

good stuff !!!!!!!! give more examples!!!

Post a Comment