\section*{Exercise 4 (5 points) -- Candidate who has followed the specialization course}
Numbers of the form $2^n - 1$ where $n$ is a non-zero natural number are called Mersenne numbers.
\begin{enumerate}
  \item We denote by $a$, $b$ and $c$ three non-zero natural numbers such that $\operatorname{GCD}(b\,;\,c) = 1$.

Prove, using Gauss's theorem, that: if $b$ divides $a$ and $c$ divides $a$ then the product $bc$ divides $a$.
  \item We consider the Mersenne number $2^{33} - 1$.

A student uses his calculator and obtains the results below:
\begin{center}
\begin{tabular}{|l|l|}
\hline
$\left(2^{33}-1\right) \div 3$ & 2863311530 \\
$\left(2^{33}-1\right) \div 4$ & 2147483648 \\
$\left(2^{33}-1\right) \div 12$ & 715827882.6 \\
\hline
\end{tabular}
\end{center}
He claims that 3 divides $\left(2^{33}-1\right)$ and 4 divides $\left(2^{33}-1\right)$ and 12 does not divide $\left(2^{33}-1\right)$.\\
  a. How does this claim contradict the result proved in question 1?\\
  b. Justify that, in reality, 4 does not divide $\left(2^{33}-1\right)$.\\
  c. By noting that $2 \equiv -1\ [3]$, show that, in reality, 3 does not divide $2^{33}-1$.\\
  d. Calculate the sum $S = 1 + 2^3 + \left(2^3\right)^2 + \left(2^3\right)^3 + \cdots + \left(2^3\right)^{10}$.\\
  e. Deduce that 7 divides $2^{33}-1$.
  \item We consider the Mersenne number $2^7 - 1$. Is it prime? Justify.
  \item We are given the following algorithm where $\operatorname{MOD}(N,k)$ represents the remainder of the Euclidean division of $N$ by $k$:
\begin{center}
\begin{tabular}{|l|}
\hline
Variables: \\
$n$ natural number greater than or equal to 3 \\
$k$ natural number greater than or equal to 2 \\
Initialization: \\
Ask the user for the value of $n$. \\
Assign to $k$ the value 2. \\
Processing: \\
While $\operatorname{MOD}\left(2^n - 1,\,k\right) \neq 0$ and $k \leqslant \sqrt{2^n - 1}$ \\
\quad Assign to $k$ the value $k+1$ \\
End While. \\
Output: \\
Otherwise \\
Display ``CASE 2'' \\
End If \\
\hline
\end{tabular}
\end{center}
  a. What does this algorithm display if we enter $n = 33$? And if we enter $n = 7$?\\
  b. What does CASE 2 correspond to?
\end{enumerate}