\(\text{{A} .................Any power of 0,1 and 5,6 end in digit itself. }\\ \text{{B}........Any ODD power of 4 end in 4 and , EVEN power end in 6.}\\\text{...............Any ODD power of 9 end in 9 and , EVEN power end in 1. }\\\text{{C}......Any power of 2 end in 2,4,8,6..................................Any power of 3 end in 3,9,7,1 .}\\ \text{............Any power of 8 end in 8,4,2,6..................................Any power of 7 end in 7,9,3,1. }\\~~\\\color{blue}{ \text{For digits X= 2,8,3,7....you may observe that its period is 4.}}\\ \)

\(For~~X^n ,~~~n~\equiv ~r~~ (mod~~ 4)\\\)

\(For ~~\color{red}{ODD}~~ powers:-\\ All~ end~ in ~ X ~~~itself~~ for~ r=1, ~~~~and~ in~~ (10-X) ~ for~ r=3. \)

\( For ~~\color{green}{EVEN} ~~powers:- \\2,~8,~ end~ in ~~4~ ~for~ r=2, ~~~~and ~in~~ 6 ~~for~ r=0\\3,~7~ end~ in~~9~~for~ r=2, ~~~and~ in~~ 1 ~~for ~r=0. \)

\(7^{95}...95\equiv 3~ (mod~~ 4).... ~~ r=3 ~ so ~~7^{95}~ end~in ~itself,~ that~ is~ 7~ at~its~ unit~ place.\\8^{1024}.... 1024\equiv 0~ (mod~~ 4).... ~~r=0~~so~~ 8^{1024} ~has~6~at~its~unit~place. \) \( \text{ I have analyzed the data taken from the web.}\)

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## Comments

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TopNewestDo you know how to prove these statements?

Note that "extensive utility of compliment of 10" is not generally true. We see this break down for the case of powers of 6. – Calvin Lin Staff · 2 years, 1 month ago

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– Niranjan Khanderia · 2 years, 1 month ago

I have changed my presentation . My only aim was to be useful to the members. As I have said before, I do not know much on number theory, I have never studied it. Even the mention of compliment of 10 is to help to remember. As far as this short note, we can see that 2 and 8, 3 and 7 behave in similar manner so one can remember easily. If you feel the note is not useful I have no objection in removing it. I welcome your suggestions. My only aim is to be useful. Not of projecting myself. I welcome your suggestions.Log in to reply

I think this note is useful, because you are pointing out a pattern that you recognized. Looking at the last digit of a power is certainly something that people do pretty often. In fact, we have an entire skill devoted to it: Finding the last digit of a power.

It is alright if you do not know why the pattern is happening, as that allows others the opportunity to explain, which will help you (and the rest of the community) understand this much better.

Not every pattern necessarily has an explanation. But for those which do, that would help us understand the underlying mathematics, and perhaps allow us to generalize even further. – Calvin Lin Staff · 2 years, 1 month ago

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