\section*{Part A}
We consider the equation (E): $15 x - 26 k = m$ where $x$ and $k$ denote relative integers and $m$ is a non-zero integer parameter.

\begin{enumerate}
  \item Justify, by stating a theorem, that there exists a pair of relative integers $( u ; v )$ such that $15 u - 26 v = 1$.\\
Find such a pair.
  \item Deduce a particular solution ( $x _ { 0 } ; k _ { 0 }$ ) of equation (E).
  \item Show that ( $x ; k$ ) is a solution of equation (E) if and only if $15 \left( x - x _ { 0 } \right) - 26 \left( k - k _ { 0 } \right) = 0$.
  \item Show that the solutions of equation (E) are exactly the pairs ( $x$; $k$ ) of relative integers such that:
$$\left\{ \begin{array} { l } 
x = 26 q + 7 m \\
k = 15 q + 4 m
\end{array} \text { where } q \in \mathbb { Z } . \right.$$
\end{enumerate}

\section*{Part B}
We associate with each letter of the alphabet an integer as indicated in the table below.

\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | c | c | c | }
\hline
A & B & C & D & E & F & G & H & I & J & K & L & M \\
\hline
0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline\hline
N & O & P & Q & R & S & T & U & V & W & X & Y & Z \\
\hline
13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 \\
\hline
\end{tabular}
\end{center}

We define an encoding system:
\begin{itemize}
  \item to each letter of the alphabet, we associate the corresponding integer $x$,
  \item we then associate to $x$ the integer $y$ which is the remainder of the Euclidean division of $15 x + 7$ by 26,
  \item we associate to $y$ the corresponding letter.
\end{itemize}

Thus, by this method, the letter E is associated with 4, 4 is transformed into 15 and 15 corresponds to the letter P, and therefore the letter E is encoded by the letter P.

\begin{enumerate}
  \item Encode the word MATHS.
  \item Let $x$ be the number associated with a letter of the alphabet using the initial table and $y$ be the remainder of the Euclidean division of $15 x + 7$ by 26.\\
a. Show then that there exists a relative integer $k$ such that $15 x - 26 k = y - 7$.\\
b. Deduce that $x \equiv 7 y + 3 ( \bmod 26 )$.\\
c. Deduce a description of the decoding system associated with the encoding system considered.
  \item Explain why the letter W in an encoded message will be decoded by the letter B.\\
Decode the word WHL.
  \item Show that, by this encoding system, two different letters are encoded by two different letters.
\end{enumerate}