An easy question that seems hard

My friend Culver Kwan asks me a question that he suddenly think of which is pretty hard when he thinks at first. However, it has a easy solution. I want to share this problem. The question is like that:

Does $n|\phi\left({m}^{n}-1\right)$ where $m, n$ are positive integers and $m>1$ ? Prove or give a counter example.

Notation meaning: $\phi\left(n\right)$ is Euler’s totient function means that the number of positive integers less than $n$ that is co-prime with $n$

#NumberTheory

Easy Math Editor

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`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`	$\sin \theta$
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Comments

It's clear that $n$ is the multiplicative order of $m$ modulo $m^n-1$ ; that is, it's the smallest positive integer $x$ such that $m^x \equiv 1 \pmod{m^n-1}.$

And $m^{\phi(m^n-1)} \equiv 1 \pmod{m^n-1}$ by Euler's theorem. So $n$ must divide $\phi(m^n-1)$ by the standard division algorithm argument.

Patrick Corn - 1 year, 9 months ago

Yeah that's my solution

Isaac YIU Math Studio - 1 year, 9 months ago

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Ella Pankhurst - 1 year, 3 months ago

This question literally seems hard but as you said it may be easy when you attempt this. But it can take a lot of time that's why I prefer online essay help for this kind of question and assignment! You can also take a look at the service so that you can explore new ways to solve these kind of issues.

victor bradford - 1 year, 2 months ago

Math	Appears 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}`	$\boxed{123}$