a) $6000$ can be factored as $2^4\times 3\times 5^3$, meaning it has $(4+1)(1+1)(3+1)=\boxed{40}$ factors.
b) $6000$ is only a multiple of $20^2$, so its square factors must be factors of $20$ squared. $20$ has $6$ factors, so there are $40-6=\boxed{34}$ nonsquare factors of $6000$.
$6000=2^4\cdot3\cdot5^3$, which has $5\cdot2\cdot4=\boxed{40}$ divisors.
We will find the number that are perfect squares. This is equal to the number of divisors of $2^2\cdot5$, which is $3\cdot2=6$.
So the answer is $40-6=\boxed{34}$ (using part A).