How many [positive](https://artofproblemsolving.com/wiki/index.php/Positive) [perfect squares](https://artofproblemsolving.com/wiki/index.php/Perfect_square) less than $10^6$ are [multiples](https://artofproblemsolving.com/wiki/index.php/Multiple) of $24$?
The [prime factorization](https://artofproblemsolving.com/wiki/index.php/Prime_factorization) of $24$ is $2^3\cdot3$. Thus, each square must have at least $3$ factors of $2$ and $1$ factor of $3$ and its square root must have $2$ factors of $2$ and $1$ factor of $3$.
This means that each square is in the form $(12c)^2$, where $12 c$ is a positive integer less than $\sqrt{10^6}$. There are $\left\lfloor \frac{1000}{12}\right\rfloor = \boxed{083}$ solutions.