Numbers that have exactly 6 positive divisors - squares are of form $p^{10}\times t$ or $p^4\times q^2 \times t$ where $p,q$ are different primes and $t$ is a s square-free integer. $2^{10}>1000$ therefore solutions are: $2^4\times 3^2 \times \{1,2,3,5,6,7\}$ $2^4\times 5^2 \times \{1,2\}$ $2^4\times 7^2$ $3^4\times 2^2 \times \{1,2,3\}$

31²<1000<32² The positive integers less than or equal to 31 that have exactly 6 divisors are 2²x3, 2x3², 2²x5, and 2²x7. ---->144,288,432,720,864,324,648,972,400,800,784 Thus, the answer is 11.