Find the smallest positive integer divisible by $10$, $11$, and $12$.
To find the smallest positive integer divisible by $10$, $11$, and $12$, I need to find the least common multiple of these three numbers. One way to do this is to prime factorize each number and then take the highest power of each prime factor that appears. So, $10 = 2 \times 5$, $11 = 11$, and $12 = 2^2 \times 3$. The highest power of $2$ that appears is $2^2$, the highest power of $3$ is $3$, the highest power of $5$ is $5$, and the highest power of $11$ is $11$. Therefore, the least common multiple of $10$, $11$, and $12$ is $2^2 \times 3 \times 5 \times 11 = 4 \times 3 \times 5 \times 11 = 60 \times 11 = 660$. So, the smallest positive integer divisible by $10$, $11$, and $12$ is $660$. 