jmc

$91$ is the first number that makes it hard to tell whether it's prime or not. Anything smaller, you can check if it's even, or ends in a $5$, or the digits sum to $3$, or maybe it's the same digit repeated twice, such as $77$. Remember that $91$ is not prime!

$87$ factors as $3\cdot 29$, $89$ is prime, $91$ factors into $7\cdot 13$, and $93$ as $3\cdot 31$. All together, that is $3^2\cdot 7\cdot 13\cdot 29\cdot 31\cdot 89$, for a total of $\boxed{6}$ different prime factors.