8.02c Divisibility by primes: algorithmic tests for primes less than 50

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OCR Further Additional Pure 2022 June Q4
9 marks Challenging +1.2
4 Let \(N\) be the number 15824578 .
    1. Use a standard divisibility test to show that \(N\) is a multiple of 11 .
    2. 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 \(\mathrm { M } = \mathrm { N } ^ { 2 }\).
        1. Express \(N\) in the unique form 101a + b for positive integers \(a\) and \(b\), with \(0 \leqslant b < 101\).
        2. Hence write \(M\) in the form \(\mathrm { M } \equiv \mathrm { r } ( \bmod 101 )\), where \(0 < r < 101\).
        3. Deduce the order of \(N\) modulo 101.
OCR Further Additional Pure AS 2018 March Q1
4 marks Moderate -0.8
1 Use standard divisibility tests to show that the number $$N = 91039173588$$
  • is divisible by 9
  • is divisible by 11
  • is not divisible by 8 .