# Unit places in power of digits 2 to 9.

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

$For~~X^n ,~~~n~\equiv ~r~~ (mod~~ 4)\$/extract_itex] $For ~~\color{#D61F06}{ODD}~~ powers:-\\ All~ end~ in ~ X ~~~itself~~ for~ r=1, ~~~~and~ in~~ (10-X) ~ for~ r=3.$ $For ~~\color{#20A900}{EVEN} ~~powers:- \\2,~8,~ end~ in ~~4~ ~for~ r=2, ~~~~and ~in~~ 6 ~~for~ r=0\\3,~7~ end~ in~~9~~for~ r=2, ~~~and~ in~~ 1 ~~for ~r=0.$ $7^{95}...95\equiv 3~ (mod~~ 4).... ~~ r=3 ~ so ~~7^{95}~ end~in ~itself,~ that~ is~ 7~ at~its~ unit~ place.\\8^{1024}.... 1024\equiv 0~ (mod~~ 4).... ~~r=0~~so~~ 8^{1024} ~has~6~at~its~unit~place.$ $\text{ I have analyzed the data taken from the web.}$  . Note by Niranjan Khanderia 5 years, 7 months ago This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science. When posting on Brilliant: • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused . • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone. • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge. • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events. MarkdownAppears as *italics* or _italics_ italics **bold** or __bold__ bold - bulleted- list • bulleted • list 1. numbered2. list 1. numbered 2. list Note: you must add a full line of space before and after lists for them to show up correctly paragraph 1paragraph 2 paragraph 1 paragraph 2 [example link](https://brilliant.org)example link > This is a quote This is a quote  # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" MathAppears as Remember to wrap math in $$ ... $$ or \[ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

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Do 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.

Staff - 5 years, 7 months 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.

- 5 years, 7 months ago

I think the previous presentation where the numbers were listed in order is easier to read and understand at a glance.

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.

Staff - 5 years, 7 months ago