HOW TO FIND THE LAST DIGIT OF ANY NUMBER

How to find the last digit of a number?

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 mⁿ
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 mⁿ 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 mⁿ 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 =3³ +7¹ -9⁴ =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

Hence the last digit is 8¹ (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
2²ⁿ ≡ 6(mod 10) when n is even

≡ 4(mod 10) when n is odd

f)3²ⁿ ≡ 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
7²ⁿ ≡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

8²ⁿ ≡ 6(mod 10) when n is even.

≡ 4(mod 10) when n is odd.

j)9ⁿ ≡ 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)5ⁿ≡5(mod 10) b)6ⁿ =6(mod 10)

c)4ⁿ ≡ 6(mod 10) when n is even .

≡ 4(mod 10) when n is odd

d)9ⁿ ≡ 1(mod 10) when n is even

≡ 9(mod 10) when n is odd

e)3²ⁿ ≡ 1(mod 10) when n is even .

≡ 9(mod 10) when n is odd

f)2²ⁿ ≡ 6(mod 10) when n is even

≡ 4(mod 10) when n is odd

g)8²ⁿ ≡ 6(mod 10) when n is even.

≡ 4(mod 10) when n is odd.

h) 7²ⁿ ≡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 5ⁿ is 5 ; the last digit in 6ⁿ is 6.

The last digit in 3⁴ⁿ is 1; The last digit in 9ⁿ is 1 or 9 according as n is odd or even. The last digit of 11ⁿ is 1, 6ⁿ is 6 , 4ⁿ is 4 or 6 according as n is odd or even.

examples:

Unit digit of 17¹⁸⁹ =unit digit of 7¹⁸⁹ =7

Unit digit of 28²⁰⁰ =unit digit of 8²⁰⁰ =6

Unit digit of 43²⁰ =unit digit of 3²⁰ =1

Unit digit of 79²⁷ =unit digit of 9²⁷ =9

Unit digit of 84³⁷ =unit digit of 4³⁷ =4