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:

185^{563} = 5; 271^{6987} = 1; 156^{25369} = 6; 190^{654789321} = 0.

Have a look at how the powers of 4 operate:

4^{1} = 4, 4^{2 }= 1__6__, 4^{3} = 6__4__ 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 9^{1} = 9, 9^{2} = 8__1__, 9^{3} = 72__9__ 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. 4^{odd} = 4 and 4^{even} = 6 and likewise 9^{odd} = 9 and 9^{even} = 1.

189^{562589743} = 9 (since power is odd); 279^{698745832} = 1(since power is even);

154^{258741369} = 4 (since power is odd); 194^{65478932} = 6 (since power is even).

2^{1 }= 2, 2^{2 }= 4, 2^{3 }= 8 & 2^{4 }= 1__6__ and after that it starts repeating.

So, the cyclicity of 2 has 4 different numbers 2, 4, 8, 6.

3^{1} = 3, 3^{2 }= 9, 3^{3} = 2__7__ & 3^{4} = 8__1__ 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.

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. 7^{1} 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.