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

[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.


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