67th Romanian Mathematical Olympiad

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.