More problems in 2016 part 4. My 400 followers problem.

Number Theory Level 5

Let \({f(x)}\) be a monic cubic polynomial such that ;

\({f(1)=14}\) , \({f(3)=46}\) and \({f(5)=150}\)

Let the integral value of \(({f(8)+f(9)+f(10)})\) be \(H\)

Then evaluate : \(\displaystyle {\left ( \sum_{d\mid H}\frac{1}{\phi (d)} \right )}\) = \(\frac{p}{q}\)

where \(p\) and \(q\) are coprime positive integers, find \((p-q)-214\)

where \(\phi(q) \) denote the Euler's totient function.


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