How many integers between $100$ and $500$ are divisibly by both $6$ and $12$, but not by $9$? The numbers divisible by both $6$ and $12$ and just the numbers divisible by $12$, so:
$12 \times 9 = 108$
$12 \times 41 = 492$
$\frac {492-108}{12} = 32 + 1 = 33$ numbers divisible by $12$, and in turn div. by $6$
$\frac {495-108}{9}+1 = 44$ numbers divisible by $9$
So there are $500-44 = 466$ numbers not divisible by $9$
Here is where I am stuck.  Any help?

The number of integers divisible by $12$ between $[100,500]$  is $$N_1=\left\lfloor\frac{500}{12}\right\rfloor-\left\lfloor\frac{100-1}{12}\right\rfloor$$
As lcm$(12,9)=18,$  among these $N_1$ numbers, the number of integers that are also divisible by $9$ between $[100,500]$ will be $$N_2=\left\lfloor\frac{500}{18}\right\rfloor-\left\lfloor\frac{100-1}{18}\right\rfloor$$
So, the required number of integers will be $N_1-N_2$
