# Unit Digit

As we have seen the cyclicity of remainders above, cyclicity exists for unit digit of the numbers also. (But there is no relation between the cyclicity of remainders and unit digit.)^{5}= 32, and so we know that unit digit of 2

^{5}is 2. But problem occurs when we start getting big numbers like 25678

^{2345}etc. To find out the unit digit of these kinds of numbers, we have some standard results, which we use as formulae.

^{4n}= â€¦6

It means that any even number raised to any power, which is a multiple of 4, will give us 6 as the unit digit.

^{4n}= â€¦1

It means that any odd number raised to any power, which is a multiple of 4, will give us 1 as the unit digit.

**0, 1, 5, 6 [these are independent of power, and unit digit will be the same respectively]**

*Exception*Example-1

Find the unit digit of 25678

^{2345}Ã— 3485^{4857}.Solution

Unit digit of 25678

^{2345}= Unit digit of 8^{45}(To find out unit digit, we need to have unit digits only. And similarly, to find out tens digit we need to have the tens and units digit only. In the present case, we are considering only last two digits of the power because divisibility rule of 4 needs only the last two digits of the number.)

8

^{45}= 8^{44+1 }**=**8^{44}Ã— 8^{1 }**=**(â€¦6) Ã— 8 = â€¦8Example-2

What is the unit digit of

_{32}32^{32}?Solution

2 is an even number which is having a power of the form 4

*n*. So, it will give 6 as the unit digit.Example-3

When 3

^{32}is divided by 50, it gives a number of the format (asdfâ€¦Â·*xy*) (*xy*being the last two digits after decimal). Find*y*.Solution

It can be observed that unit digit of 3

^{32}= 1. Now any number having 1 as the unit digit will always give 2 at the unit place when divided by 50.So, answer is 2.

Example-4

What is the last non-zero digit of the number 30

^{2720}?**(CAT 2005, 2 marks)**Solution

30

^{2720}= [30^{4}]^{680 }= â€¦10000 .... 00Unit digit can also be found out by cyclicity method as well.

It can be seen that

Unit digit of 2

^{1}= 2Unit digit of 2

^{2}= 4Unit digit of 2

^{3}= 8Unit digit of 2

^{4}= 6Unit digit of 2

^{5}= 2So, it can be inferred that unit digit of 2

^{1}= unit digit of 2^{5}= unit digit of 2^{9}Cyclicity of 3 = 4

Cyclicity of 4 = 2

Cyclicity of 7 = 4

Cyclicity of 8 = 4

Cyclicity of 9 = 2

Cyclicity of 0 or cyclicity of 1 or cyclicity of 5 or cyclicity of 6 = 1. To know more about Unit Digid, refer to the book. Demystifying Number System.