4 Let \(N\) be the number 15824578 .
- Use a standard divisibility test to show that \(N\) is a multiple of 11 .
- 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 .
- Justify the validity of this process.
- Continue the student's test to show that \(7 \mid N\).
(iii) Given that \(N = 11 \times 1438598\), explain why 7| 1438598 . - Let \(\mathrm { M } = \mathrm { N } ^ { 2 }\).
- Express \(N\) in the unique form 101a + b for positive integers \(a\) and \(b\), with \(0 \leqslant b < 101\).
- Hence write \(M\) in the form \(\mathrm { M } \equiv \mathrm { r } ( \bmod 101 )\), where \(0 < r < 101\).
- Deduce the order of \(N\) modulo 101.