Example 3 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 Mathematics Competition)
Construct the subset $A$ as follows:
Notice that $1995 \div 15=133$, and let
$$
A_{1}=\{134,135, \cdots, 1995\} \text {, }
$$

then $\left|A_{1}\right|=1862$.
Also, $133=15 \times 8+13$, and let
$A_{2}=\{9,10, \cdots, 133\}$,

then $\left|A_{2}\right|=125$.
Let $A_{3}=\{1,2, \cdots, 8\}$, then $\left|A_{3}\right|=8$.
Thus, $M=A_{1} \cup A_{2} \cup A_{3}, A_{i} \cap A_{j}=\varnothing(i \neq j)$.
When $a \in A_{1}$, $15 a \notin A$; when $a \in A_{2}$, at least one of $a$ and $15 a$ does not belong to $A$; when $a \in A_{3}$, $15 a \notin A_{1}$. Therefore, let $A=A_{1} \cup A_{3}$, then $A$ is a subset that meets the requirements, and
$$
|A|=\left|A_{1}\right|+\left|A_{3}\right|=1862+8=1870 .
$$