\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
  1. Show that for every natural integer \(a\): $$a^{31} - a \equiv 0 \pmod{62}$$
  2. Show that for all natural integers \(a\) and \(n\) with \(n \geq 1\): $$a^{30+n} - a^n \equiv 0 \pmod{62}$$

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