Let $a, b$ and $n$ be natural numbers. Prove that if $a^n$ and $b^n$ are relatively prime, then $a$ and $b$ are relatively prime.

Question

Let $a, b$ and $n$ be natural numbers. Prove that if $a^n$ and $b^n$ are relatively prime, then $a$ and $b$ are relatively prime.

I have been able to prove the above statement by contrapositive in the following way: If $a$ and $b$ are not relatively prime, then $a^n$ and $b^n$ are not relatively prime.

Suppose $a$ and $b$ are not relatively prime. Then $\exists d \in \mathbb{N}$ such that $ d \vert a \wedge d \vert b $ where $d>1$.

Let $n \in \mathbb{N}$.

Since $$a^n = \overbrace{a\cdot a \dots \cdot a}^{n\text{-times}}$$ and $$b^n = \overbrace{b\cdot b \dots \cdot b}^{n\text{-times}}$$ then $$d \vert a^n \wedge d \vert b^n$$ which just implies that $a^n$ and $b^n$ are not relatively prime.

Could someone also show the direct proof?

Answer 1

Since $\gcd(a^n,b^n)=1,\exists x,y\in\Bbb Z|a^nx+b^ny=1\implies a\cdot(a^{n-1}x)+b\cdot(b^{n-1}y)=1$.

Thus, $\exists p=a^{n-1}x,q=b^{n-1}y$, both integers, such that $ap+bq=1\implies\gcd(a,b)=1$.

Answer 2

Hint: If a prime $p$ divides the product $x=ab$, then it divides a factor, i.e., $p\mid a$ or $p\mid b$.

Answer 3

$\,\cal P(n) := $ set of prime factors of $\,n.\,$ By unique factorization, $\,\cal \color{#c00}{P(a^k) = \cal P(a)}\,$ for $\,k\ge 1,\,$ so

$$\gcd(a,b)\!=\!1\!\iff\! \cal P(a)\cap \cal P(b) = \emptyset\color{#c00}\!\iff\! P(a^m)\cap \cal P(b^n) =\emptyset \!\iff\! \gcd(a^m,b^n)\! =\! \rm1,\ \,m,n\ge 1 $$