An association assigns each registered child a 6-digit number $c_1 c_2 c_3 c_4 c_5 k$ where:
\begin{itemize}
  \item $c_1 c_2$ represents the last two digits of the child's birth year;
  \item $c_3 c_4 c_5$ are three digits chosen by the association;
  \item $k$ is a check digit computed as follows:
\end{itemize}

\begin{itemize}
  \item we compute the sum $S = c_1 + c_3 + c_5 + a \times (c_2 + c_4)$ where $a$ is an integer between 1 and 9;
  \item we perform the Euclidean division of $S$ by 10, the remainder obtained is the check digit $k$.
\end{itemize}

When an employee enters the 6-digit number of a child, an input error can be detected when the sixth digit is not equal to the check digit calculated from the first five digits.

\begin{enumerate}
  \item In this question only, we choose $a = 3$.\\
a. Can the number 111383 be that of a child registered with the association?\\
b. The employee, confusing a brother and sister, exchanges their birth years: 2008 and 2011. Thus, the number $08c_3c_4c_5k$ is transformed into $11c_3c_4c_5k$. Is this error detected thanks to the check digit?
  \item We denote $c_1c_2c_3c_4c_5k$ the number of a child. We seek the values of the integer $a$ for which the check digit systematically detects the typing error when the digits $c_3$ and $c_4$ are swapped. We therefore assume that the digits $c_3$ and $c_4$ are distinct.\\
a. Show that the check digit does not detect the error of swapping the digits $c_3$ and $c_4$ if and only if $(a-1)(c_4 - c_3)$ is congruent to 0 modulo 10.\\
b. Determine the integers $n$ between 0 and 9 for which there exists an integer $p$ between 1 and 9 such that $np \equiv 0 \pmod{10}$.\\
c. Deduce the values of the integer $a$ which allow, thanks to the check digit, to systematically detect the swap of the digits $c_3$ and $c_4$.
\end{enumerate}