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 .

The answer is 249.

1 solution

Otto Bretscher
Dec 31, 2015

Just a brief outline of a solution on this busy last day of the year.

We can solve a system of three linear equations to find $f(x)=10+3x+x^3$ , with $H=f(8)+f(9)+f(10)=2352=3*7^2*2^4$ . Now $g(n)=\sum_{d|n}\frac{1}{\phi(d)}$ is a multiplicative function, so that $g(2352)=g(3)g(7^2)g(2^4)=\left(1+\frac{1}{3-1}\right)\left(1+\frac{1}{7-1}+\frac{1}{49-7}\right)\left(1+1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}\right)=\frac{575}{112}$ . The answer is $\boxed{249}$

Happy 2016 everybody!

Great mapping of the solution .

A Former Brilliant Member - 5 years, 5 months ago

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

The answer is 249.

1 solution

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