I don't know how to count

Algebra Level 4

Consider a quintic function \(f(x)\) such that \(f(k) = k \) for \(k=1,2,3,4,5\). Given that \(f(x) = g(x) + q(x) \) such that \(g(x)\) has all real roots and its leading coefficient is an integer and \(q(x) \) is a linear function, which of these answer choices is \((f(6) - 6)\) always divisible by?

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