Find the number of positive integers $n$ less than or equal to $100$, which are divisible by $3$ but are not divisible by $2$.

Step 1: The number of positive integers less than or equal to $100$ that are divisible by $3$ is given by:

Step 2: The numbers divisible by $3$ and also divisible by $2$ are those divisible by $\text{lcm}(2,3)=6$. The count is:

Step 3: The numbers divisible by $3$ but not by $2$ is the difference:

These are the odd multiples of $3$ up to $100$: $\{3,9,15,\dots ,99\}$, which forms an arithmetic progression with common difference $12$.