Straight squares

Number of perfect Squares Between $n$ and $m$ = $\lfloor(\sqrt{m})\rfloor-\lceil(\sqrt{n})\rceil-1$ , if m and n are both perfect squares

Number of perfect Squares Between $n$ and $m$ = $\lfloor(\sqrt{m})\rfloor-\lceil(\sqrt{n})\rceil$ , if m or n is/not a perfect square

Number of perfect Squares Between $n$ and $m$ = $\lfloor(\sqrt{m})\rfloor-\lceil(\sqrt{n})\rceil+1$ , if m and n are not perfect squares

Number of perfect Squares Between 1 and 1000 = $\lfloor(\sqrt{1000})\rfloor\ = 31$

Number of Perfect Squares Divisible by 3 = $\lfloor\dfrac{31}{3}\rfloor\ = 10$

All multiples of 3, when squared, are divisible by 3. Therefore, given that 32 squared > 1000, all the multiples of 3 from 1 - 32, when squared, is divisible by 3. The answer, therefore, is 10 .