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Let $G=\mathbf{Z}/18\mathbf{Z}\times \mathbf{Z}/60\mathbf{Z}$ and consider the group homomorphism $f\colon G\to G\colon x\mapsto 4x$. In other words, $f(x)=x+x+x+x$.

Let $f^k$ denote the $k$-th composite of $f$ with itself, where $f^1=f$. How can I find the smallest $k\ge2$ such that $f^k=f$?

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