1. Find the number of positive integers among the first $10^{5}$ that are not divisible by $7, 11, 13$.
1. Solution: According to Theorem 1 of this chapter, the number of integers sought is
$$\begin{array}{c}
10^{5}-\left[\frac{10^{5}}{7}\right]-\left[\frac{10^{5}}{11}\right]-\left[\frac{10^{5}}{13}\right]+\left[\frac{10^{5}}{7 \times 11}\right] \\
+\left[\frac{10^{5}}{7 \times 13}\right]+\left[\frac{10^{5}}{11 \times 13}\right]-\left[\frac{10^{5}}{7 \times 11 \times 13}\right]=10^{5} \\
-14285-9090-7692+1298+1098+699-99=71929
\end{array}$$