The Noisy Max mechanism and its variations are fundamental private selection algorithms that are used to select items from a set of candidates (such as the most common diseases in a population), while controlling the privacy leakage in the underlying data. A recently proposed extension, Noisy Top-k with Gap, provides numerical information about how much better the selected items are compared to the non-selected items (e.g., how much more common are the selected diseases). This extra information comes at no privacy cost but crucially relies on infinite precision for the privacy guarantees. In this paper, we provide a finite-precision secure implementation of this algorithm that takes advantage of integer arithmetic.