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Apologies that the title is vague, but I’m not quite sure where this question would fall under modular arithmetic. The question is:

Let $k$ be a natural number with hcf$(k,p-1)=1$ where $p$ is prime. Prove that every integer has a $k^{th}$ root modulo p.

I really don’t even know where to begin with this so could someone please give me an indication of how to start the proof and then hopefully from there I should be able to finish the problem.

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closed as off-topic by Hurkyl, Namaste, Saad, The Phenotype, Claude Leibovici Mar 25 '18 at 7:54

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  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Hurkyl, Namaste, Saad, The Phenotype, Claude Leibovici
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Oh yes, sorry for missing that detail I’ll add it to the question :) $\endgroup$ – MichealAlex456 Mar 24 '18 at 13:13
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There exists $r,s \in \Bbb Z$ such that $kr + (p-1)s = 1$. By Fermat's Little Theorem,

$$a \equiv a^{kr + (p-1)s} \equiv (a^r)^k \pmod p.$$

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