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This question already has an answer here:

I don't know how to calculate the following modulo: $$321^{654} \mod 1013$$

Are there some easy way to do this?

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marked as duplicate by Watson, Davide Giraudo, Winther, Jean-Claude Arbaut, JMP Sep 26 '16 at 0:15

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Working in the prime field $\Bbb F_{1013}$ we can use equalities.

There are several ways to solve according the easier we can. For example

$$321^{654}=(321^{109})^6$$ so from $321^{10}=52$ we have $$321^{100}=52^{10}=781\Rightarrow321^{109}=781\cdot(321)^9=185\cdot1013+35$$ It follows $$321^{654}=35^6=\color{red}{863}$$

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