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Find a primitive root modulo each of the following integers.

a) $13$

My TA said we are not going to go over this. We did not go over the topic. It seems like something good to know though.

Is there anyone who can help me with this?

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Just go through all elements and check whether they generate or not. When you find one that generates everything you've found a primitive element. – Ittay Weiss Apr 30 at 19:59
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Check the definition of a primitive root. Then try numbers from $1$ to $12$ at random (or systematically, your call!) and check whether they are primitive roots. Nothing to it but put your shoulder to the wheel! – Jyrki Lahtonen Apr 30 at 19:59
If you want to do some proper thinking...$13$ is prime...does what you found generalise? – user1729 Apr 30 at 20:08

marked as duplicate by TMM, Amzoti, Davide Giraudo, Lord_Farin, rschwieb Apr 30 at 21:24

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From Wikipedia:

... a primitive root modulo $n$ is any number $g$ with the property that any number coprime to $n$ is congruent to a power of $g$ modulo $n$. In other words, $g$ is a generator of the multiplicative group of integers modulo $n$. That is, for every integer $a$ coprime to $n$, there is an integer $k$ such that $g^k \equiv a\; (mod \;n)$.

So you want to find an integer $g$ such that when you take powers of $g$, then you can get all numbers from $1$ to $12$ (mod $13$). (Since $13$ is prime). So just pick a number $g$ and compute $$ g, g^2, g^3, \dots. \quad(\text{mod } 13) $$ If you find a $g$ such that this list contains all numbers $1,\dots, 12$, then you have found a primitive root.

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