Is it true that $\forall x,y,n\in \mathbb{Z}$, if $\gcd(x,y)=1$ then $\gcd(x^n, y)=1$? If not, is there a counterexample?
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It's true, and it's a very elementary thing to prove depending on what tools you have. Unless you tell us about what you have available, that's pretty much all that can be said. – Apr 06 '19 at 00:18
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The answer to both questions is "yes". – Apr 06 '19 at 00:23
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@T.Bongers how about only from the definition of gcd? – qweruiop Apr 06 '19 at 00:23
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4Duplicate of Prove that if $\gcd(a,b)=1$ then $\gcd(a^m, b)=1$ – Bill Dubuque Apr 06 '19 at 00:24
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Also a dupe of this, where there are many other approaches (please don't duplicate these or the prior-linked answers below). – Bill Dubuque Apr 06 '19 at 01:11
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By Bezout there is $u,v$ so that $xu+yv=1$. Using binomial theorem yields: $$(xu+yv)^n=x^n{u^n}+ yv\sum_{k=0}^{n- 1}{{n\choose k} (ux)^k(vy)^{n-k-1}}=1$$ Now putting $U’=u^n$ and $V’= v\sum_{k=0}^{n- 1}{{n\choose k} (ux)^k(vy)^{n-k-1}}$ gives: $$U’x^n+V’y=1.$$Hence $\gcd(x^n,y)=1$
DINEDINE
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@qweruiop In fact the answer is a dupe of an answer in the dupe target I posted above. Please don't duplicate prior answers, esp. if they are already linked on the question. – Bill Dubuque Apr 06 '19 at 01:05
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@BillDubuque This is not a dupe, since the guy who provide this answer hasn’t make a look into your dupe link. – DINEDINE Apr 06 '19 at 01:09
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1@BillDubuque: it is quite possible that this answer was in progress when you posted your link. And my guess is that this question will be closed and deleted soon. – robjohn Apr 06 '19 at 01:16
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@robjohn Possibly, but in any case this approach using binomial Bezout (or ideal) expansion (for same or mixed powers) probably occurs in at least 50 prior answers if not more. There is no good reason to further such duplication. – Bill Dubuque Apr 06 '19 at 01:20
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Hint: Assume $\gcd(x^n,y)\neq 1$. Then there exists a prime $p$ that divides both terms...
Carl Schildkraut
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1I wonder why there are downvotes on this. I don't really see an issue with this, unless I'm overlooking something. – PrincessEev Apr 06 '19 at 00:29
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@EeveeTrainer A guess: the question is a FAQ that has been asked and answered tens (if not hundreds) of times already. Unless one has something significantly novel to add then it is best to not further such duplication since it is bad for the health of the site. – Bill Dubuque Apr 06 '19 at 00:57