Question 7: A number $n$ is a perfect square if $\sqrt{n}$ is an integer. If we factor out $n$ as a product of primes $n = p_1^{e_1}\cdot p_2^{e_2}\cdots p_k^{e_k}$ for $n$ to have an integer square root we would need to be able to split all the factors in two equal groups, thus we need for each of the exponents $e_i$ to be even: $\sqrt{n} = p_1^{e_1/2}\cdot p_2^{e_2/2}\cdots p_k^{e_k/2}$. In this particular problem we want to find perfect squares that divide $2593 = 2^5 \cdot 3^4$. Any factor of this number will be of the form $2^a \times 3^b$ with $a = 0,1,2,3,4,5$ and $b = 0,1,2,3,4$. Since we want the factors to be perfect squares, we need that $a$ and $b$ are both even, so that reduces our options of $a$ and $b$ to $a = 0, 2, 4$ and $b = 0, 2, 4$. Thus, we have three options to choose from for the exponent for $2$ and three options to choose from for the exponent of $3$, so there are $3 \cdot 3 = 9$ distinct perfect squares that divide $2592$.

Question 9: The problem is asking for the largest possible value of $\overline{7a4b}$, but it doesn't say that all digits are distinct, so it is possible that $a$ and $b$ have the same value.