Suppose $Z_1,...Z_n$ are i.i.d $N(\mu,\sigma^2)$ where both $\mu$ and $\sigma$ are integers, while $\sigma$ is a power of 10. (So $\log_{10}(\sigma)\in\mathbb{Z}$). I am trying to show that the MLE can be found by rounding the sample mean to the nearest integer. Additionally I want to construct an estimator for $\sigma$ that maximizes the likelihood that $\log_{10}(\sigma)\in\mathbb{Z}$.
Attempt: define the function $f(\delta)=\sum_{i=1}^n(x_i-(\bar{x}+\delta))^2$. This function is even as a function of $\delta$. The Log-likelohood is of this form. Since this is a parabola we have that the maximum must occur at the unique value (since $f$ is an even function, our max has to be unique) however I have no idea how to incorporate the fact that $\mu$ is an integer? Furthermore I have no idea how to construct an estimator for $\sigma$.