jmc

Because $24=3\cdot 2^3$, a square is divisible by 24 if and only if it is divisible by $3^2\cdot 2^4=144$. Furthermore, a perfect square $N^2$ less than $10^6$ is a multiple of 144 if and only if $N$ is a multiple of 12 less than $10^3$. Because 996 is the largest multiple of 12 less than $10^3$, there are $\frac{996}{12}=83$ such positive integers less than $10^3$ and $\boxed{83}$ positive perfect squares which are multiples of 24.