[Math] If $a$ and $b$ are relatively prime then so are $a$ and $b^n$ for every positive integer $n$

Question

I have been asked to prove the following via induction (as the textbook as suggested):

If $a$ and $b$ are relatively prime then so are $a$ and $b^n$ for every positive integer $n$

So, I did the following but I'm stuck at the induction step.

Base Case $n=1$

$(a,b^1) =(a,b)= 1$. This is true via the definition provided in the problem.

Inductive Hypothesis

Suppose, this is true for all $n$. $$(a,b^n) = 1$$

Inductive Step for $n+1$

\begin{align}
&(a,b^{n+1}) = 1 \\
&(a,b^nb) = 1 \\
\end{align}

Now, I am stuck from here. Quite honestly, I think have done the entire induction wrong. what I would have done next is convert the $\gcd$ into the algebraic form of $ka + lb^nb = 1$ but I'm not too sure.

Please provide only hints.

Thanks a lot!

Answer 1

Since you only want hints:

Try to prove the following identity: $$\gcd(a,b\cdot c) = \gcd(a,b)\cdot\gcd(a,c)$$ This identity basically finishes your proof, given your induction hypothesis.

Also, regarding your induction hypothesis, you should change "true for all $n$" to "true for all $n\leq k$". Otherwise, you are assuming what you want to prove. After that little change, the inductive step is then to prove this is true for $n = k+1$.