[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.