Example 4 Let $M=\{1,2,3, \cdots, 1995\}, A$ be a subset of $M$ and satisfy the condition: if $x \in A, 15 x \notin A$, then the maximum number of elements in $A$ is $\qquad$
(1995, National High School Mathematics Competition)
Solution: Construct subset $A$ as follows:
$1995 \div 15=133$, let $A_{1}=\{134,135, \cdots$, 1995\}, then $\left|A_{1}\right|=1862$ (elements);
$133 \div 15=8$ remainder 13, let $A_{2}=\{9,10, \cdots$, 133\}, then $\left|A_{2}\right|=125$ (elements);

Let $A_{3}=\{1,2,3,4,5,6,7,8\}$, then $\left|A_{3}\right|=$ 8 (elements).

Obviously, $\bigcup_{j=1}^{3} A_{j}=\{1,2,3, \cdots, 1995\}, A_{i} \cap$ $A_{j}=\varnothing, i \neq j$, and $\left\{15 a \mid a \in A_{i+1}\right\} \subset A_{i}, i=$ $1,2,3$.

Let $A=A_{1} \cup A_{3}$, then, $A$ is the required subset and
$$
|A|=\left|A_{1}\right|+\left|A_{3}\right|=1862+8=1870 \text {. }
$$