AMC 12 2001 Problem 21

Four positive integers $a$, $b$, $c$, and $d$ have a product of $8!$ and satisfy:

$$\begin{align*} ab + a + b & = 524 \\ bc + b + c & = 146 \\ cd + c + d & = 104 \end{align*}$$

What is $a-d$?

$\text{(A) }4 \text{(B) }6 \text{(C) }8 \text{(D) }10 \text{(E) }12$