Linear modular constraints are a powerful class of constraints that arise naturally in cryptanalysis, checksums, hash functions, and the like. Given their importance, the past few years have witnessed the design of combinatorial solvers with native support for linear modular constraints, and the availability of such solvers has led to the emergence of new applications. While there exist global constraints in CP that consider congruence classes over domain values, linear modular arithmetic constraints have yet to appear in the global constraint catalogue despite their past investigation in the context of model counting for CSPs. In this work we seek to remedy the situation by advocating the integration of linear modular constraints in state-of-the-art CP solvers. Contrary to previous belief, we conclude from an empirical investigation that Gauss-Jordan Elimination based techniques can provide an efficient and scalable way to handle linear modular constraints. On the theoretical side, we remark on the pairwise independence offered by hash functions based on linear modular constraints, and then discuss the design of hashing-based model counters for CP, supported by empirical results showing the accuracy and computational savings that can be achieved. We further demonstrate the usefulness of native support for linear modular constraints with applications to checksums and model counting.