How many positive $3$-digit numbers are multiples of $20$, but not of $55$?
A multiple of $20$ and $55$ is a multiple of their least common multiple, which is $220$. To find the number of multiples of $20$ between $100$ and $999$, I can divide the difference between the largest and smallest such multiples by $20$, and then add $1$. The smallest multiple of $20$ that is a $3$-digit number is $100$, and the largest is $980$. So the number of multiples of $20$ between $100$ and $999$ is $\frac{980 - 100}{20} + 1 = 45$. To find the number of multiples of $220$ between $100$ and $999$, I can use the same method, but with $220$ instead of $20$. The smallest multiple of $220$ that is a $3$-digit number is $220$, and the largest is $880$. So the number of multiples of $220$ between $100$ and $999$ is $\frac{880 - 220}{220} + 1 = 4$. Therefore, the number of multiples of $20$, but not of $55$, between $100$ and $999$ is $45 - 4 = 41$. 