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How would you demonstrate that the greatest common divisor of two integers, say x and y, to an arbitrary power greater than or equal to one is equal to the greatest common divisor of those same integers to the same power.

In other words, that for an integer $d \geq 1$, the $GCD(a^d, b^d) = GCD(a, b)^d$.

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Let $g=\gcd(a,b)$ and $a=a'g,b=b'g$ where $a',b'$ have no common factors.

We have

$$a^d=a'^dg^d,\\b^d=b'^dg^d.$$

Then it is clear that $g^d$ divides both $a^d$ and $b^d$. And $a'^d,b'^d$ can't have common factors.

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