Example 1 Let $M=\{1,2, \cdots, 1995\}, A$ be a subset of $M$ and satisfy the condition: when $x \in A$, $15 x \notin A$, then the maximum number of elements in $A$ is $\qquad$ (1995 National High School League Question)
To construct a subset $A$ that satisfies the conditions and contains the maximum number of elements, we need to ensure that when $x \in A$, $15x \notin A$. This is only true when $15x > 1995$, which means $x > 133$. Therefore, $A$ can include $\{134, 135, \cdots, 1995\}$. Additionally, to ensure that when $x \in A$, $15x \notin A$, we need $15x < 134$, which means $x < 9$. Hence, $A$ can also include $\{1, 2, \cdots, 8\}$. Thus, we take the subset $A = \{1, 2, \cdots, 8\} \cup \{134, 135, \cdots, 1995\}$, which satisfies the conditions. In this case, the number of elements in $A$ is $|A| = 8 + (1995 - 133) = 1870$.

On the other hand, for any subset $A$ of $M$ that satisfies the conditions, since at least one of $x$ and $15x$ (where $x = 9, 10, 11, \cdots, 132, 133$) must not belong to $A$, the number of elements in $A$ is $|A| \leq 1995 - (133 - 8) = 1870$. Therefore, the maximum number of elements in $A$ is 1870.