a) We check the numbers decreasingly: $99 = 3^2 \cdot 11$, $99$ has $6$ divisors and $6 \nmid 99$; $98 = 2 \cdot 7^2$, $98$ has $6$ divisors and $6 \nmid 98$; $97 = 97$, $97$ has $2$ divisors and $2 \nmid 97$; $96 = 2^5 \cdot 3$, $96$ has $12$ divisors and $12 \mid 96$. So $96$ is the largest exquisite two digit number.

b) Let $X$ be a positive integer with its last digit $3$.

Then $X$ is odd and $X$ is not a perfect square, and a non-perfect square has an even number of divisors. Since an odd number cannot be a multiple of an even number, $X$ cannot be exquisite.