Example 1 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
(1995 National High School League Question)
Construct the 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).
Clearly, $A_{1} \cup A_{2} \cup A_{3}=\{1,2, \cdots, 1995\}, A_{i} \cap A_{j}=\varnothing, i \neq j$, and $\{15 a \mid a \in \left.A_{i+1}\right\} \subseteq A_{i}, i=1,2,3$.
Let $A=A_{1} \cup A_{3}$, then $A$ is the required subset, and
$|A|=\left|A_{1}\right|+\left|A_{2}\right|=1862+8=1870$.