Inspired by Mehul Chaturvedi

Algebra Level 5

\[P(x)=\dfrac{ax^7}{b}+\dfrac{cx^6}{d}+\dfrac{ex^5}{f}+\dfrac{gx^4}{h}+\dfrac{ix^3}{j}+\dfrac{kx^2}{l}+\dfrac{mx}{n}-22\]

Let a polynomial \(P(x)\) be defined such that \(P(n)\) is the number of open-chain structural isomers of \(n\) carbon membered alkanes for \(n\in [1,8]\).

Given that \(a\), \(c\), \(e\), \(g\), \(i\),\(k\) and \(m\) are positive integers such that \(\gcd(a,b)=\) \(\gcd(c,d)=\) \(\gcd(e,f)=\) \( \gcd(g,h)=\) \(\gcd(i,j)=\) \(\gcd(k,l)=\) \(\gcd(m,n)= 1\).

Calculate \(a+b+c+d+e+f+g+h+i+j+k+l+m+n-22\).

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