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I am unable to prove 2 statements. Please help
1)(a+b)^n - a^n - b^n is always divisible by ab for all n belongs to N.
2) a polynomial of odd degree will always have one of its roots to be either +1 or -1.

Note by Raven Herd
1 year, 7 months ago

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For statement 1), use the binomial theorem to find that

\((a + b)^{n} = a^{n} + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^{2} + .... + \binom{n}{n-1}ab^{n-1} + b^{n}.\)

After subtracting \(a^{n}\) and \(b^{n}\) the remaining terms are all divisible by \(ab,\) and thus so is their sum.

Statement 2) is not actually true. \(f(x) = x - 2\) is of odd degree and only has root \(2.\) Brian Charlesworth · 1 year, 7 months ago

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@Brian Charlesworth sir , I have a problem . I very much want to study number theory but I am unable to grasp the concepts beyond modular multiplication . I earnestly want to learn those Fermat rules ,Euler theorem , CRT etc,etc, Please help. Raven Herd · 1 year, 7 months ago

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@Raven Herd Have you tried the wikis here on Brilliant.org? For example, here is the one on the CRT.

This can be found by choosing "Topics", followed by "Number Theory", "Modular Arithmetic" and then the 'open book' icon to the right of "Chinese Remainder Theorem". You can find the other topics in your list in this fashion as well. Brian Charlesworth · 1 year, 7 months ago

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@Brian Charlesworth yes I have tried them. I devoted full 1 week and I was quite confident that I can learn it . But it didn't work. Please reply soon. P.S. Sir, an odd request : Where is the started problems option? I am hunting it since the set up of Brilliant has changed. Raven Herd · 1 year, 7 months ago

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@Raven Herd I'm not sure what to suggest next. If you spent a week going through all the wikis then you probably know more than I do now about this topic. Sometimes it's just about practice, so keep trying Number Theory questions on Brilliant (and elsewhere) and maybe the concepts will eventually become second nature. Sorry I couldn't suggest something more helpful. :(

As for the "Started Problems" option, I noticed it missing from the main page a few days ago as well. I eventually found it, though. Click on the "blue planet" icon and choose the "View mobile site" option. Once on that page, click on the "three dot" icon in the upper right corner and you will see the "Started problems" option listed there. Then, if you want, you can choose the "View full site" option on the same list to take your list of started problems back to the main site format. Brian Charlesworth · 1 year, 7 months ago

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@Brian Charlesworth Sir ,you are too modest.I have decided that I will go through the wikis again.I believe I have not been sincere and Brilliant wikis are the best study material I have come across till now and the best part about them is that they are prepared by people who find a particular topic fascinating and thus eliminate the chances of errors greatly. Anyways, thanks for you consideration. Raven Herd · 1 year, 7 months ago

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@Raven Herd You're welcome. Good luck with your studies. :) Brian Charlesworth · 1 year, 7 months ago

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