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Proving that a number has a multiplicative inverse

Is there a way to prove, without using calculator, that $a^n-1$ has a multiplicative inverse?
If I start with $a^n-1=b$ I would have to prove that $b$ is not the additive inverse of $a^n-1$ and that there is not $c$ such that $a^n-1=bc$.
But how can I prove that $b$ is not the additive inverse of $a^n-1$ without the calculator?
Is there a way to prove that there is not $c$ such that $a^n-1=bc$ without using the calculator?


Let $a$ be a fixed natural number.
If $n$ is odd then we have
a^{n}-1 = \underbrace{a^n}_{\in\mathbb{N}}-1 = (a-1)(a^{n-1}+\cdots+a+1)
\gcd(a^n-1,a-1) = \gcd(a^n,a-1)=a-1.
If $n$ is even then
a^{n}-1 = (a-1) \cdot a^{n-1}+\cdots+a+1,
\gcd(a^n-1,a-1) = \gcd(a^n,a-1)=a-1.
Thus, in either case we have $\gcd(a^n-1,a-1)=a-1$.
Since $\gcd(a^n-1,a-1)=a-1$, it follows that there exists $b$ such that $b\cdot a^n-1=1$, and there exists $c$ such that $c\cdot (a^n-1)=b$.
Now, $a^n-1=b$ is equivalent to $a^n=1+b$ and therefore $a$ is a root of the polynomial $x^n-1$.

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KEYMACRO is a utility that provides a simple way to record your keystrokes and save them in a text file that you can use later to automate tasks.
KeyMACRO Keystroke Monitoring Software
Designed to work with Microsoft Windows operating systems, KEYMACRO Keystroke Monitoring Software monitors your computer’s keyboard and saves every single keystroke you perform. After this, you can later review the recorded data and use it to automate certain actions. You can perform the following operations:
Record keystrokes for later use
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Examples of Automation
To illustrate the efficiency of macros created with KEYMACRO Keystroke Monitoring Software, we’ve given some examples of how you can use macros to perform various tasks. You can create macros and save them in a text file to create different kinds of automation scenarios:
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By saving a series of keystrokes, you can create a specific script that you can later use to automate any task. The script simply opens a folder or folder and saves the current date and time in a specific location. After that, you can later on perform a task every day or any other time interval.
Automate tasks using macros


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