This is my attempt to solve the following question: "Use induction to show that if $(a,b)=1$ (greatest common divisor of $a$ and $b$), then $(a,b^n)=1$ for all $n\geq 1$."
We have that $(a,b)=1$, which implies that $1=au + bv$ for some integers $u,v$. For the induction, assume that $(a,b^k)=1$, which implies $1=ar + b^{k}s$ for some integers $r,s$. Multiplying these equations together yields $1=(au + bv)(ar + b^{k}s)$. We then get
$$1=a^{2}ur + aub^{k}s + bvar + b^{k+1}vs$$ which simplifies to $$1=a(aur + ub^{k}s + bvr) + b^{k+1}vs$$.
$(a,b^{k+1})$ divides the right side of the equation, which implies that $(a,b^{k+1})$ also divides the left side of the equation, and since $(x,y)$ of any integers $x$ and $y$ is at least $1$, it follows that $(a,b^{k+1})=1$. This completes the proof by induction, since we have assumed our base case of $n=1$.
I would greatly appreciate any feedback if any of the steps in the solution are wrong. Thanks in advance.
This solution is correct, very clear, and easy to follow. Well done!
The only downside is that this question is very easily generalisable and your proof method is very specific.