The Smallest \(k\)

Number Theory

Let \[f(x)=7x^{11}+11x^{7}+9kx\] \[g(x)=7x^{13}+13x^{7}+10kx \]

Find the minimum positive value of \(k\) for which \(77| f(x)\) and \(91|g(x)\) for every integer \(x\)

Details and Assumptions

We use "\(\mid\)" to mean "divides evenly into". For example, \(3\mid6\), and \(12\mid132\).