Finding the number of distinct common roots of unity

Question

Consider equations:

$${x}^{p} = 1\\ {x}^{q} = 1$$

Then the number of common roots is equal to the $\gcd(p,q).$

But, how can I prove this statement?

Answer 1

WLOG, assume that $p > q$.
From the division algorithm, we know that $p = aq+b$. Then,

$$x^p = x^{aq+b} = (x^q)^ax^b = x^b = x^{p\text{ mod }q}.$$

Here, we have condensed the problem from common roots of $x^p$ and $x^q$ to the common roots of $x^p$ and $x^{p\text{ mod }q}$.

Can you see a correspondence to the Euclidean algorithm?