8.02o Binomial theorem: (a+b)^p = a^p + b^p (mod p) for prime p

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OCR Further Additional Pure 2018 September Q5
11 marks Hard +2.3
  1. You are given that \(N = \binom{p-1}{r}\), where \(p\) is a prime number and \(r\) is an integer such that \(1 \leq r \leq p - 1\). By considering the number \(N \times r!\), prove that \(N \equiv (-1)^r \pmod{p}\). [5]
  2. You are given that \(M = \binom{2p}{p}\), where \(p\) is an odd prime number. Prove that \(M \equiv 2 \pmod{p}\). [6]