4 Let $N$ be the number 15824578 .
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Use a standard divisibility test to show that $N$ is a multiple of 11 .
\item A student uses the following test for divisibility by 7 .

\begin{displayquote}
'Throw away' multiples of 7 that appear either individually or within a pair of consecutive digits of the test number.\\
Stop when the number obtained is $0,1,2,3,4,5$ or 6 .\\
The test number is only divisible by 7 if that obtained number is 0 .
\end{displayquote}

For example, for the number $N$, they first 'throw away' the " 7 " in the tens column, leaving the number $N _ { 1 } = 15824508$. At the second stage, they 'throw away' the " 14 " from the left-hand pair of digits of $N _ { 1 }$, leaving $N _ { 2 } = 01824508$; and so on, until a number is obtained which is $0,1,2,3,4,5$ or 6 .

\begin{itemize}
\end{enumerate}\item Justify the validity of this process.
  \item Continue the student's test to show that $7 \mid N$.\\
(iii) Given that $N = 11 \times 1438598$, explain why 7| 1438598 .
\item Let $\mathrm { M } = \mathrm { N } ^ { 2 }$.
\begin{enumerate}[label=(\roman*)]
\item Express $N$ in the unique form 101a + b for positive integers $a$ and $b$, with $0 \leqslant b < 101$.
\item Hence write $M$ in the form $\mathrm { M } \equiv \mathrm { r } ( \bmod 101 )$, where $0 < r < 101$.
\item Deduce the order of $N$ modulo 101.
\end{itemize}
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{OCR Further Additional Pure 2022 Q4 [9]}}