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3)Given a prime $p$ and an integer $a$, we say that $a$ is a $\textit{primitive root} \pmod p$ if the set $\{a,a^2,a^3,\ldots,a^{p-1}\}$ contains exactly one element congruent to each of $1,2,3,\ldots,p-1\pmod p$.

For example, $2$ is a primitive root $\pmod 5$ because $\{2,2^2,2^3,2^4\}\equiv \{2,4,3,1\}\pmod 5$, and this list contains every residue from $1$ to $4$ exactly once.

However, $4$ is not a primitive root $\pmod 5$ because $\{4,4^2,4^3,4^4\}\equiv\{4,1,4,1\}\pmod 5$, and this list does not contain every residue from $1$ to $4$ exactly once.

What is the sum of all integers in the set $\{1,2,3,4,5,6\}$ that are primitive roots $\pmod 7$?

I have no clue how to answer this question. Any help will be great.

Thank you very much.

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    $\begingroup$ This looks like three questions to me. They would be better posed separately. Also MathJax would be useful in questions 1 and 2 too. $\endgroup$ – Lord Shark the Unknown Aug 30 '19 at 4:01
  • $\begingroup$ Welcome to Mathematics Stack Exchange. For question 3, you could see whether each of the integers in the set is a primitive root and add up the ones that are $\endgroup$ – J. W. Tanner Aug 30 '19 at 4:05
  • $\begingroup$ Find the primitive roots mod $7$, then add them up. $\endgroup$ – runway44 Aug 30 '19 at 4:09
  • $\begingroup$ @runway44 How would I find the primitive roots of mod 7. Thank you very much for your help. $\endgroup$ – a23 Aug 30 '19 at 4:12
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    $\begingroup$ Use the definition you gave for primitive roots $\endgroup$ – J. W. Tanner Aug 30 '19 at 4:14
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Modulo $7$, the powers of $1$ are $1,1,1,1,1,1,$ so $1$ is not a primitive root; the powers of $2$ are $2,4,1,2,4,1,$ so $2$ is not a primitive root; and the powers of $3$ are $3,2,6,4,5,1$, so $3$ is a primitive root. Can you take it from here? Test whether $4,5,$ and $6$ are primitive roots, and then compute the sum of the primitive roots as requested.

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