> 
> If the sequence $10^n\bmod m$ (where n>=0) is purely periodic, show that
>  $\gcd(m,10) = 1$
> 
> 
> 

I know that if the sequence is purely periodic, $a\_i=a\_{i+P}$ where P is the period. I can show using examples that the gcd(m,10) must be 1 but not generally

I know that if the gcd(m, 10) = 1

10x + my = 1 

or

10x $\equiv$ 1 mod m 

But from here I'm not sure how to prove the gcd must be 1