Need help proving the relation above, I'm assuming it can be done using gcd properties.
2026-04-06 15:08:22.1775488102
Show that $\forall{a,b,n}\in\mathbb{N},\ gcd(a^n,b^n)=gcd(a,b)^n$.
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Write $a=gx$ and $b=gy$ where $g=gcd(a,b)$. Then $x,y$ are relatively prime. So $a^n = g^nx^n$ and $b=g^ny^n$. Since $x^n$ and $y^n$ are relatively prime we have $$gcd(a^n,b^n) = g^n$$