Positive natural numbers $≤ 200$ divisible by one and only one between $12$ and $15$

Question

Calculate how many positive natural numbers $≤ 200$ are divisible by one and only one between $12$ and $15$.

My attempt:

$B_1=[200/12]=16$

$B_2=[200/15]=13$

$B_1∩B_2=3$

$B_1+B_2+2|B1∩B2|=16+13+6=35$

Answer 1

Let $A$ be the number of positive integers $\leq 200$ divisible by $12$ and let $B$ be the number of positive integers $\leq 200$ divisible by $15$. Then we are after $A\Delta B=(A\setminus B)\cup (B\setminus A)$ (the symmetric difference). But $$ |A\Delta B|=|A|+|B|-2|A\cap B|\tag{1}. $$ Note that $|A|=[200/12]$ and $B=[200/15]$ where $[\cdot]$ is the floor. An integer is divisible by $12$ and $15$ iff it is divisible by the least common multiple, namely, $60$. Hence $|A\cap B|=[200/60]$.