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I have the following problem that I can't seem to solve in closed form. I would like to minimize $$ m\left\lceil\frac{2k}{m}+x\right\rceil-2k$$ subject to $m\geq m_0$ and $\mathop{\mathrm{gcd}}(k,m)=1$, $0<k<m$, where $x\in\mathbb{R}$ and $k,m\in\mathbb{Z}$. where $x$ is a given real number. If there are multiple minimizers, I would like the one with the highest value of $m$.

Is there a technique for this? The approach of asking for the fractional part $2k/m+x$ to be as close to $1$ as possible doesn't seem to work because I also need $k$ to have no common factors with $m$.

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