Powers in finite fields

Task number: 2482

For \(n\in \mathbb N\) and an asociative operation \(\cdot\) let \(a^n=a\cdot a\cdot \ldots \cdot a\), where the element \(a\) appears \(n\) times in the product.

  • Variant 1

    Determine values \(2^{101},3^{1\,001}\) and \(4^{1\,000\,001}\) in the field \(\mathbb Z_{17}\).

  • Variant 2

    Determine \(5^{100},8^{200},11^{300}\) and \(18^{400}\) in the field \(\mathbb Z_{19}\).

Difficulty level: Easy task (using definitions and simple reasoning)
Solution require uncommon idea
Cs translation
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