2.3 Euler's Theorem

Euler's Theorem


  1. What is the value of \(\Phi(175) \), where \(\Phi \) is Euler's Totient function?
    Noting that \(175 = 5^2 \cdot 7\). It follows that \(\Phi(175)=(5^2 - 5^1)(7-1) = 20 \cdot 6 = 120\).
  2. What is the remainder of \(22^{12001}\) divided by \(175\)?
    \(22^{12001} = 22^{(120 \cdot 100) + 1} = 22^{120^{100}} \cdot 22 \equiv 1^{100} \cdot 22 \equiv 22 \text{ mod } 175\).