Fermat's Little Theorem

Questions requiring application of Fermat's Little Theorem to evaluate large powers modulo a prime.

4 questions · Challenging +1.2

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OCR Further Additional Pure Specimen Q9
14 marks Hard +2.3
9
  1. (a) Prove that \(p \equiv \pm 1 ( \bmod 6 )\) for all primes \(p > 3\).
    (b) Hence or otherwise prove that \(p ^ { 2 } - 1 \equiv 0 ( \bmod 24 )\) for all primes \(p > 3\).
  2. Given that \(p\) is an odd prime, determine the residue of \(2 ^ { p ^ { 2 } - 1 }\) modulo \(p\).
  3. Let \(p\) and \(q\) be distinct primes greater than 3 . Prove that \(p ^ { q - 1 } + q ^ { p - 1 } \equiv 1 ( \bmod p q )\). \section*{END OF QUESTION PAPER} {www.ocr.org.uk}) after the live examination series.
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Edexcel FP2 2019 June Q4
12 marks Standard +0.3
    1. Use Fermat's Little Theorem to find the least positive residue of \(6 ^ { 542 }\) modulo 13
    2. Seven students, Alan, Brenda, Charles, Devindra, Enid, Felix and Graham, are attending a concert and will sit in a particular row of 7 seats. Find the number of ways they can be seated if
      1. there are no restrictions where they sit in the row,
    3. Alan, Enid, Felix and Graham sit together,
    4. Brenda sits at one end of the row and Graham sits at the other end of the row,
    5. Charles and Devindra do not sit together.
Edexcel FP2 2024 June Q1
4 marks Standard +0.3
  1. In this question you must show detailed reasoning.
Use Fermat's Little Theorem to determine the least positive residue of
\(21 { } ^ { 80 } ( \bmod 23 )\)
(4)
OCR Further Additional Pure 2018 December Q3
9 marks Challenging +1.8
3
  1. Show that \(10 ^ { 2 } \equiv 6 ( \bmod 47 )\).
  2. Determine the integer \(r\), with \(0 < r < 47\), such that \(6 r \equiv 1 ( \bmod 47 )\).
  3. Determine the least positive integer \(n\) for which \(10 ^ { n } \equiv 1\) or \(- 1 ( \bmod 47 )\).