Example 1 From the 205 positive integers $1,2, \cdots, 205$, what is the maximum number of integers that can be selected such that for any three selected numbers $a, b, c (a<b<c)$, we have
$$
a b \neq c ?^{[1]}
$$
(2005, (Casio Cup) National Junior High School Mathematics Competition)
Estimate first.
Since $14 \times 15=210>205$, then $14,15, \cdots$, 205 satisfy that for any three numbers $a 、 b 、 c(a<b<c)$, we have $a b \neq c$.
Because 1 multiplied by any number equals the number itself, $1,14,15, \cdots, 205$ satisfy the condition.
Therefore, there are $205-14+1+1=193$ numbers in total.
If we select one more number from $2 \sim 13$, what would happen?
Construct the following 12 sets:
$(2,25,2 \times 25),(3,24,3 \times 24), \cdots$,
$(13,14,13 \times 14)$.
The 36 numbers in the above sets are all distinct, and the smallest number is 2, while the largest number is $13 \times 14=182<205$.

Therefore, all three numbers in each set cannot be selected.
In summary, the number of numbers that satisfy the given condition does not exceed $205-12=193$.