Exercise 0.3.13 (Invertible elements)

Question

Let $n \in \mathbb{Z}, n>1$ and let $a \in \mathbb{Z}$ with $1\leq a \leq n$. Prove that if $a$ and $n$ are relatively prime then there is an integer $c$ such that $ac \equiv 1 \pmod n$ [use the fact that the g.c.d. of two integers is a $\mathbb{Z}$-linear combination of the integers].

Richard Ganaye · Accepted Answer

(Already done in Exercise 9.)
\begin{proof} 
If $d = a \wedge n = 1$, there are integers $c$ and $e$ such that $1 = ca  + e n$. Then $ac \equiv 1 \pmod n$.

\end{proof}

Exercise 0.3.13 (Invertible elements)

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Exercise 0.3.13 (Invertible elements)

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