Great The Number of Divisor X

Note first that $\dfrac{x^{3} - 7}{x - 7} = x^{2} + 7x + 49 + \dfrac{336}{x - 7}.$

So $x - 7 | x^{3} - 7$ whenever $x - 7$ is a divisor of $336.$

Now $336 = 2^{4}*3*7,$ and so has $(4 + 1)(1 + 1)(1 + 1) = 20$ positive divisors and thus $40$ integer divisors in total. Each of these divisors corresponds to a distinct integer $x$ such that $x - 7 | x^{3} - 7,$ and so the desired answer is $\boxed{40}.$