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Prove the following:

  1. $U_n=\{Z \in \mathbb{C} : Z^n = 1\}$. Then, show that $U_a \cap U_b = U_{gcd(a,b)}$
  2. Hence prove that, $gcd(x^a-1,x^b-1)=x^{gcd(a,b)}-1$

I started out with if $α \in U_a$, then $α = e^{\frac{2ikπ}a}$ where $0<k<a$ and if $β \in U_b$, then $β = e^{\frac{2ilπ}b}$ where $0<l<b$. So, if $α=β$, then $e^{\frac{2ikπ}a}=e^{\frac{2ilπ}b}$ $\implies \frac kl=\frac ba$

Then i claimed that : For every $k, 0<k<gcd(a,b)$, we find $l$ such that $\frac kl=\frac ba$

But then I am not able to prove my claim. Please help me.

Gloona
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  • [This post](https://math.stackexchange.com/questions/7473/prove-that-gcdan-1-am-1-a-gcdn-m-1?noredirect=1&lq=1) has really many answers. I suppose there is one, which you can use. – Dietrich Burde May 05 '22 at 14:59
  • I am not getting an answer by using the first part and that is what I want. – Gloona May 05 '22 at 15:07
  • The first part is the above duplicate, see [here](https://math.stackexchange.com/questions/2453328/for-alpha-in-mathbbt-prove-that-alpham-1-and-alphan-1-only-if?rq=1). – Dietrich Burde May 05 '22 at 16:10
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    See the "alternative" argument using "order" in [this answer](https://math.stackexchange.com/a/7561/242) in the linked dupe. – Bill Dubuque May 05 '22 at 16:13

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