2.3: Problem 10: Find the number of positive integers \(\leq 3600\) that are prime to \(3600\).
Solution
The prime factorization of \(3600\) is as follows
$$
\begin{align*}
3600 = 2^4 \cdot 3^2 \cdot 5^2
\end{align*}
$$
We can use Euler’s Totient function to compute the number of positive numbers that are prime to \(3600\) as follows
$$
\begin{align*}
\phi(3600) &= 3600 \cdot \left( 1- \frac{1}{2}\right) \cdot \left(1- \frac{1}{3}\right) \cdot \left(1- \frac{1}{5} \right) \\
&= 3600 \cdot \frac{1}{2} \cdot \frac{2}{3} \cdot \frac{4}{5} \\
&= 1800 \cdot \frac{2}{3} \cdot \frac{4}{5} = 1200 \cdot \frac{4}{5} = 960 \\
\end{align*}
$$
Therefore, we have \(960\) positive integers less than \(3600\) that are prime to \(3600\).