So if there is a question which says for eg Find the unit digit of 7^24, then why we use the unit digit of 7^4 series only to get the unit digit, why not 7^2 series or why not 7^1 series, even these both powers are divisible by 24. So why would we use only 7^4? I hope you guys understood my Question.
Anything works. Turns out that using 7^4 makes calculating the units digit of 7^{24} much easier.
Note that
7^{24} = (7^4)^6
Since the units digit of 7^4 is 1, then the units digit of 7^{24} is simply 1^6 = 1.
Had you used 7^2 (whose units digit is 9) instead, then to find the units digit of 7^{24} = (7^2)^{12}, you would be faced with the subproblem of finding the units digit 9^{12}, which is arguably no easier than the original problem at hand.
But with this logic, to find the unit digit of a no. we can just use the unit no. to the power 1 as 1 is divisible by every no. Eg 23^14, wud follow 3^1 series , 7^45 would follow 7^1 series, etc.
How does that help?
Carrying out our “algorithm”, we get:
7^{45} = (7^1)^{45}
The units digit of 7^1 is 7, and so the units digit of 7^{45} is (7^1)^{45}.
The whole process achieved nothing other than just restating the question.
I would carefully reread what I mentioned prior just so that we’re on the same page.