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Let $p, q$ be distinct odd primes, I would like to compute the number of solution $\mod p$ to the congruence $x^q \equiv 1 \mod p$. (Show that it's $gcd(q, p - 1))$

If $x^q \equiv 1 \mod p$, since $q$ is prime, this must mean that $x$ is either $1$ or has order $q$. Hence, the problem reduces to finding the number of elements of $U_p$ which is either $1$ or has order $q$. If $ q \nmid p - 1$, then there is only one element, namely $1$. If $q \mid p - 1$, then the number of elements of order $q$ is $\phi(q) = q - 1$, and so the number of solutions to the congruence $\mod p$ is $q - 1 + 1 = q$, and in either case the number of solutions is $\gcd(q, p - 1)$.

Does this look fine?

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    $\begingroup$ Yes, it sounds fine, given you have already done the theorems implicitly quoted. A fairly simple argument using a primitive root of $p$ (generator of the multiplicative group) proves the same result for $x^k\equiv 1\pmod{p}$, where $k$ is not necessarily prime. $\endgroup$ Mar 5, 2013 at 1:04

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The key to questions like this is that (1) the group of nonzero elements of $\mathbb Z/(p)$ is cyclic of order $p-1$; and (2) in a cyclic group of order $n$, there is just one subgroup of order $d$ for every divisor $d$ of $n$, and these are the only subgroups of the original cyclic group. Now you fill in the details.

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