Exercise 2.3.5 (Number of generators for $\mathbb{Z}/49000\, \mathbb{Z}$.)

Question

Find the number of generators for $\mathbb{Z}/49000\, \mathbb{Z}$.

Richard Ganaye · Accepted Answer

ewcommand{\Z}{\mathbb{Z}}

\begin{proof} 
The number of generators for $\Z/49000\, \Z$ is
\begin{align*}
\varphi(49000) &= \varphi(2^3\cdot 5^3 \cdot 7^2)\
&=(2^3 - 2^2)(5^3 - 5^2)(7^2 - 7)\
&=4 \cdot 100\cdot 42\
&= 16800.
\end{align*}

\end{proof}
\begin{verbatim}
sage: euler_phi(49000)
16800
\end{verbatim}

Exercise 2.3.5 (Number of generators for $\mathbb{Z}/49000\, \mathbb{Z}$.)

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Exercise 2.3.5 (Number of generators for $\mathbb{Z}/49000\, \mathbb{Z}$.)

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