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# Finding the unit digit in case of the numbers when one number is raised to the power of other

[latexpage]
[unit digit of odd number, unit digit of even number, Number System, product of unit digit]
We can categories every number raised to certain power into two parts :

(I) When odd Number is raised to some power.

i.e. \${21}^{24}\$ , \${23}^{26}\$ ,\${25}^{234}\$ ,\${27}^{78}\$ ,\${29}^{67}\$

(II) When Even Number is raised to some power.

i.e. \${20}^{24}\$ , \${22}^{26}\$ ,\${24}^{234}\$ ,\${26}^{78}\$ ,\${28}^{67}\$

(I) When Odd Number is raised to some power:

In this case , Lets multiply the unit digit of the number itself until we get 1 as the unit digit. We find a pattern in this.

\${(….1)}^{4n}\$ = (….1)

\${(….3)}^{4n}\$ = (….1)

\${(….7)}^{4n}\$ = (….1)

\${(….9)}^{4n}\$ = (….1)

i.e.any odd number(except numbers ending with 5) multiplied with itself 4 times or multiple of 4 times , It would end with 1 as unit digit.

Odd Numbers with 5 as unit digit would always end with 5.

Q.)Find out the unit digit of \${23}^{26}\$ .

\${23}^{26}\$ = \${23}^{24}\$ X \${23}^{2}\$

Now, here 24 is a multiple of 4 , hence the unit digit of \${23}^{24}\$ will be 1 .

unit digit of \${23}^{2}\$ = 9

Hence , Unit digit of \${23}^{26}\$ = unit digit of \${23}^{24}\$ X unit digit of \${23}^{2}\$ =unit digit of(1 X 9) = 9

Similarly , We can find out the unit digit of the other odd numbers raised to some powers.

In the case of 5 , It’s unit digit will always be 5 no matter what .

(II) When Even Number is raised to some power:

In this case , Lets multiply the unit digit of the number itself until we get 6 as the unit digit. We find a pattern in this.

\${(….2)}^{4n}\$ = (….6)

\${(….4)}^{4n}\$ = (….6)

\${(….6)}^{4n}\$ = (….6)

\${(….8)}^{4n}\$ = (….6)

i.e.any odd number(except numbers ending with 0) multiplied with itself 4 times or multiple of 4 times , It would end with 6 as unit digit.

Even Numbers with 0 unit digit would always end with 0 irrespective of powers.

e.g. Find the unit digit of \${28}^{203}\$

\${28}^{203}\$ = \${28}^{200}\$ X \${28}^{3}\$

Now, 200 is a multiple of 4 , hence \${28}^{203}\$ would end with unit digit as 6.

unit digit of \${28}^{3}\$ = unit digit of (8X8X8) = 2

Hence, unit digit of \${28}^{203}\$ = unit digit of \${28}^{200}\$ X unit digit of \${28}^{3}\$ = unit digit of (6X2) = 2

Similarly we can mix odd and even numbers and can find out the unit digit of the product using the separate logic for even and odd.