OCR Further Additional Pure AS 2023 June — Question 5

Exam BoardOCR
ModuleFurther Additional Pure AS (Further Additional Pure AS)
Year2023
SessionJune
TopicNumber Theory
5
  1. Express as a decimal (base-10) number the base-23 number \(7119 _ { 23 }\).
  2. Solve the linear congruence \(7 n + 11 \equiv 9 ( \bmod 23 )\).
  3. Let \(N = 10 a + b\) and \(M = a + 7 b\), where \(a\) and \(b\) are integers and \(0 \leqslant b \leqslant 9\).
    1. By considering \(3 N - 7 M\), prove that \(23 \mid N\) if and only if \(23 \mid M\).
    2. Use a procedure based on this result to show that \(N = 711965\) is a multiple of 23 .
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