8.02k Euclid's lemma: if a|rs and hcf(r,a)=1 then a|s

1 questions

Sort by: Default | Easiest first | Hardest first
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.
    If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
    OCR is part of the