# Primitive Root Modular Arithmetic Question

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.

• This looks like three questions to me. They would be better posed separately. Also MathJax would be useful in questions 1 and 2 too. Aug 30, 2019 at 4:01
• 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 Aug 30, 2019 at 4:05
• Find the primitive roots mod $7$, then add them up. Aug 30, 2019 at 4:09
• @runway44 How would I find the primitive roots of mod 7. Thank you very much for your help.
– a23
Aug 30, 2019 at 4:12
• Use the definition you gave for primitive roots Aug 30, 2019 at 4:14

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.