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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.)
Taking a very simple example– 25 = 32, and so we know that unit digit of 25 is 2. But problem occurs when we start getting big numbers like 256782345 etc. To find out the unit digit of these kinds of numbers, we have some standard results, which we use as formulae.
(Any even number)4n = …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.
(Any odd number)4n = …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.
Exception 0, 1, 5, 6 [these are independent of power, and unit digit will be the same respectively]
Find the unit digit of 256782345 × 34854857.
Unit digit of 256782345 = Unit digit of 845
(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.)
845 = 844+1 844 × 81 (…6) × 8 = …8
What is the unit digit of 323232?
2 is an even number which is having a power of the form 4n. So, it will give 6 as the unit digit.
When 332 is divided by 50, it gives a number of the format (asdf…·xy) (xy being the last two digits after decimal). Find y.
It can be observed that unit digit of 332 = 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.
What is the last non-zero digit of the number 302720(CAT 2005, 2 marks)
302720 = [304]680 = …10000 .... 00
Unit digit can also be found out by cyclicity method as well.
It can be seen that
Unit digit of 21 = 2
Unit digit of 22 = 4
Unit digit of 23 = 8
Unit digit of 24 = 6
Unit digit of 25 = 2
So, it can be inferred that unit digit of 21 = unit digit of 25 = unit digit of 29
Hence, cyclicity of 2 = 4, i.e., every fourth power of 2 will give the same unit digit.
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.

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