GCD ... period!

For m>1m>1, it can be proven that the integer sequence fm(n)=gcd(n+m,mn+1)f_m(n) = \gcd(n+m,mn+1) has a fundamental period Tm.T_m. In other words, nN, fm(n+Tm)=fm(n).\forall n \in \mathbb{N}, \space f_m(n+T_m) = f_m(n). Find an expression for TmT_m in terms of m,m, and then give your answer as T12.T_{12}.

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