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The question is: Show for all n >= 1, that [tex]10^n[/tex] leaves remainder 1 after dividing by 9.

So I said that there is an integer m, such that [tex]10^n = 9m + 1[/tex]

I also see that m has to be 1, or 11, or 111, or 1111, ..... but I am unsure of where to go from there and if that is even useful.

So how I am supposed to use induction here? I would think that I would want m in terms of n, but I am not seeing it. Any ideas would be great, thanks!