1. The number of simplest proper fractions with 2088 as the denominator is
$\qquad$.


II.1.672.
Since $2088=2^{3} \times 3^{2} \times 29$, the number of integers from $1,2, \cdots$, 2088 that contain at least one of the factors $2,3,29$ is
$$
\begin{array}{l}
\left(\left[\frac{2088}{2}\right]+\left[\frac{2088}{3}\right]+\left[\frac{2008}{29}\right]\right)- \\
\left(\left[\frac{2088}{2 \times 3}\right]+\left[\frac{2088}{2 \times 29}\right]+\left[\frac{2088}{3 \times 29}\right]\right)+ \\
{\left[\frac{2088}{2 \times 3 \times 29}\right]} \\
=1416 \text {. } \\
\end{array}
$$

Therefore, the number of integers from 1 to 2088 that are coprime with 2088 is $2088-1416=672$.