# GCD ... period!

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

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