It's all about number theory! (1)

{ab+a+b=524bc+b+c=146cd+c+d=104\begin{cases} ab+a+b = 524\\ bc+b+c = 146 \\ cd+c+d = 104 \end{cases}

The product of four positive integers aa, bb, cc, and dd is 8!8! . If these numbers satisfy the system of equations above, find the value of ada-d.


See also It's all about number theory! (2).
This problem is not original. It is taken from one of the problems at a local olympiad in Medellín, Colombia.
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