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Math Help - Find the number of primitive roots in Z_101, Z_12

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    Find the number of primitive roots in Z_101, Z_12

    Find the number of primitive roots in \mathbb{Z}_{101}, \mathbb{Z}_{12}.


    I cannot find an example from my lecture notes about primitive roots in \mathbb{Z}_n. What is the definition of primitive roots in \mathbb{Z}_n?

    The answers are 40 and none respectively.
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    Re: Find the number of primitive roots in Z_101, Z_12

    A primitive root in \mathbb Z_n^\times is an integer k such that for each i\in\{1,2,\ldots,n-1\} such that \gcd(i,n)=1, there exists an integer m such that i\equiv k^m\mod n. (It follows that \gcd(k,n)=1.)

    When n=p is a prime, \mathbb Z_p^\times always has primitive roots. Indeed \mathbb Z_p^\times is a cyclic group of order p-1 generated by any primitive root; hence the number of primitive roots is \varphi(p-1). This answers your question for \mathbb Z_{101}^\times.

    If n is not prime, things are a little complicated. For n=12, though, you can easily see that k^2\equiv1\mod{12} for k=1,5,7,11, which are the integers coprime with 12. Hence \mathbb Z_{12}^\times has no primitive roots since there is are no integers k,m such that k^m\equiv5\mod{12} even though \gcd(5,12)=1.
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    Re: Find the number of primitive roots in Z_101, Z_12

    Thank you. I see. This is really the same as primitive roots in \mathbb{U}_{101} and \mathbb{U}_{12}, just in different notation.
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