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A few hints to get you started:Alexis87 said:I have encountered difficulties in solving this question. Your help is greatly appreciated!
I think the basis for an induction proof that $(a,b^n)=1$ for every positive integer $n$ is $(a,b)=1$ and not $(a,b^2)=1$.Opalg said:That equation is of the form $aX + b^2Y = 1$, which shows that $(a,b^2) = 1.$
Can you use that as the basis for an inductive proof that $(a,b^n) = 1$ for all $n$?
Yes, of course. But I was not implying that the base case should be $(a,b^2)=1$. What I was suggesting was a technique that might be used for proving the inductive step.Evgeny.Makarov said:I think the basis for an induction proof that $(a,b^n)=1$ for every positive integer $n$ is $(a,b)=1$ and not $(a,b^2)=1$.
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