(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 \(p > 3\).
Given that \(p\) is an odd prime, determine the residue of \(2 ^ { p ^ { 2 } - 1 }\) modulo \(p\).
Let \(p\) and \(q\) be distinct primes greater than 3 . Prove that \(p ^ { q - 1 } + q ^ { p - 1 } \equiv 1 ( \bmod p q )\).
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