The concept that revolves around finding the unit digit of a number uses the basics of number system. Learning this concept means you have strengthened your basic concepts.
The concept of unit digit can be learned by figuring out the unit digits of all the single digit numbers from 0 - 9 when raised to certain powers. The first learning in that for you will be that these numbers can be broadly classified into three categories for this purpose:
Digits 0, 1, 5 & 6: When we observe the behavior of these digits, they all have the same unit’s digit as the number itself when raised to any power, i.e. 0n = 0, 1n =1, 5n = 5, 6n = 6. So, it becomes simple to understand this logic.
Example: Finding the Unit digit of following numbers:
185563 = 5; 2716987 = 1; 15625369 = 6; 190654789321 = 0.
Digits 4 & 9: Both these numbers are perfect squares and also have the same behavior with respect to their unit digits i.e. they have a cyclicity of only two different digits as their unit’s digit.
Have a look at how the powers of 4 operate:
41 = 4, 42 = 16, 43 = 64 and so on.
Hence, the power cycle of 4 contains only 2 numbers 4 & 6, which appear in case of odd and even powers respectively.
Likewise 91 = 9, 92 = 81, 93 = 729 and so on.
Hence, the power cycle of 9 also contains only 2 numbers 9 & 1, which appear in case of odd and even powers respectively.
So broadly these can be remembered in even and odd only, i.e. 4odd = 4 and 4even = 6 and likewise 9odd = 9 and 9even = 1.
Example: Finding the Unit digit of following numbers:
189562589743 = 9 (since power is odd); 279698745832 = 1(since power is even);
154258741369 = 4 (since power is odd); 19465478932 = 6 (since power is even).
Digits 2, 3, 7 & 8: These numbers have a power cycle of 4 different numbers.
21 = 2, 22 = 4, 23 = 8 & 24 = 16 and after that it starts repeating.
So, the cyclicity of 2 has 4 different numbers 2, 4, 8, 6.
31 = 3, 32 = 9, 33 = 27 & 34 = 81 and after that it starts repeating.
So, the cyclicity of 3 has 4 different numbers 3, 9, 7, 1.
7 and 8 follow similar logic.
So these four digits i.e. 2, 3, 7 and 8 have a unit digit cyclicity of four steps.
To summarize, we can say that since the power cycle of these numbers has 4 different digits, we can divide the power by 4, find the remaining power and calculate the unit’s digit using that.
Example: Find the Unit digit of 287562581
The first observation for this question: the unit digit involved is 7, which has a four step cycle. You need to divide the power by 4 and obtain the remaining power. Doing so, you get the result as 1. Now the last step is to find the unit’s digit in this power of the base i.e. 71 has the unit’s digit as 7, which will become the answer.
The above set of examples explains how you the concept of cyclicity to obtain the unit digit of numbers. In case you understood the above examples, you should be easily able to obtain the unit digit of the numbers given above, and in fact, you should be able to extend this learning to as many examples as you want.
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