Exercise 2.3.10 (Number of positive integers $n \leq 3600$ prime to $3600$)

Question

Find the number of positive integers $\leq 3600$ that are prime to $3600$.

Richard Ganaye · Accepted Answer

\begin{proof} By definition of $\phi$, this number is $N = \phi(3600)$.
Since $3600 = 2^4\cdot 3^2 \cdot 5^2,$
\begin{align*}
\phi(3600) &= 3600 \left(1-\frac{1}{2}ight)\left(1-\frac{1}{3}ight)\left(1-\frac{1}{5}ight)\
&= 3600 \cdot \frac{1}{2} \cdot \frac{2}{3}\cdot  \frac{4}{5}\
&=960.
\end{align*}
So $N = \phi(3600) = 960$ is the number of positive integers $n \leq 3600$ that are prime to $3600$.

\bigskip
With Sage:
\begin{verbatim}
sage: euler_phi(3600)
960
\end{verbatim}
\end{proof}

Exercise 2.3.10 (Number of positive integers $n \leq 3600$ prime to $3600$)

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Exercise 2.3.10 (Number of positive integers $n \leq 3600$ prime to $3600$)

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