Exercise 2.3.3 (Generators of $\mathbb{Z}/ 48\mathbb{Z}$)

Question

Find all generators of $\mathbb{Z}/ 48\mathbb{Z}$.

Richard Ganaye · Accepted Answer

\begin{proof} 
The generators of $\mathbb{Z}/48\mathbb{Z}$ are the elements $\overline{d} \in \mathbb{Z}/48\mathbb{Z}\ (0 \leq d < 48)$ such that $d \wedge 48 = 1$ (i.e, such that $2 
mid d, 3 
mid d$). This gives
$$d \in \{ 1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47\}.$$

(As a verification $\varphi(48) = \varphi(3) \varphi(2^4) = (3 - 1)(2^4 - 2^3) = 16$, so there are $16$ generators of  $\mathbb{Z}/48\mathbb{Z}$.)

\end{proof}
With Sagemath:
\begin{verbatim}
sage: [d for d in range(48) if gcd(d,48) == 1]
[1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47]
sage: euler_phi(48)
16
\end{verbatim}

Exercise 2.3.3 (Generators of $\mathbb{Z}/ 48\mathbb{Z}$)

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Exercise 2.3.3 (Generators of $\mathbb{Z}/ 48\mathbb{Z}$)

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