Question 4
N4 (FRA). Denote by $S$ the set of all primes $p$ such that the decimal representation of $1/p$ has its fundamental period divisible by $3$. For every $p\in S$ such that $1/p$ has its fundamental period $3r$ one may write $1/p=0.a_1a_2\ldots a_{3r}a_1a_2\ldots a_{3r}\ldots$, where $r=r(p)$; for every $p\in S$ and every integer $k\ge1$ define $f(k,p)$ by
$$f(k,p)=a_k+a_{k+r(p)}+a_{k+2r(p)}.$$
(a) Prove that $S$ is infinite.
(b) Find the highest value of $f(k,p)$ for $k\ge1$ and $p\in S$.
$$f(k,p)=a_k+a_{k+r(p)}+a_{k+2r(p)}.$$
(a) Prove that $S$ is infinite.
(b) Find the highest value of $f(k,p)$ for $k\ge1$ and $p\in S$.