Great Mathematics Olympiad problem!

Number Theory Level 4

All pairs \((a. b)\) of positive integers such that \(a{b}^{2} + b + 7\) divides \({a}^{2}b+ a + b\) are given by
\((p, q)\) ; \((s, t)\) and \((\alpha, \beta)\).

Find the value of \(p + q + s + t\).

Here, \(\alpha, \beta\) varies over a variable for example it can be \((4{ a }^{ 2 }, 5a)\) for some variable \(a\).

And \(p, q, s, t\) are natural numbers.


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