\textbf{Exercise 5 — Candidates who have followed the specialization course}\\
The natural integers $1, 11, 111, 1111, \ldots$ are rep-units. These are natural integers written only with 1s.\\
For all non-zero natural integer $p$, we denote $N _ { p }$ the rep-unit written with $p$ times the digit 1:

$$N _ { p } = \underbrace { 11 \ldots 1 } _ { \substack { p \text { repetitions } \\ \text { of the digit 1 } } } = \sum _ { k = 0 } ^ { k = p - 1 } 10 ^ { k }$$

Throughout the exercise, $p$ denotes a non-zero natural integer.\\
The purpose of this exercise is to study some properties of rep-units.

\textbf{Part A: divisibility of rep-units in some particular cases}

\begin{enumerate}
  \item Show that $N _ { p }$ is divisible neither by 2 nor by 5.
  \item In this question, we study the divisibility of $N _ { p }$ by 3.\\
a. Prove that, for all natural integer $j$, $10 ^ { j } \equiv 1 \bmod 3$.\\
b. Deduce that $N _ { p } \equiv p \bmod 3$.\\
c. Determine a necessary and sufficient condition for the rep-unit $N _ { p }$ to be divisible by 3.
  \item In this question, we study the divisibility of $N _ { p }$ by 7.\\
a. Copy and complete the congruence table below, where $a$ is the unique relative integer belonging to $\{ - 3 ; - 2 ; - 1 ; 0 ; 1 ; 2 ; 3 \}$ such that $10 ^ { m } \equiv a \bmod 7$. No justification is required.

\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | }
\hline
$m$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
$a$ &  &  &  &  &  &  &  \\
\hline
\end{tabular}
\end{center}

b. Let $p$ be a non-zero natural integer. Show that $10 ^ { p } \equiv 1 \bmod 7$ if and only if $p$ is a multiple of 6. You may use the Euclidean division of $p$ by 6.\\
c. Justify that, for all natural integer $p$ non-zero, $N _ { p } = \frac { 10 ^ { p } - 1 }{ 9 }$.
\end{enumerate}