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This note presents two original and creative divisibility tests for the integers 2 and 5. Both rules use the factorial of the last digit of a number, combined with a small correction when the factorial value would disrupt the required modular congruence. The resulting tests are mathematically equivalent to the classical last-digit rules but reach the same conclusion through an unconventional factorial-based approach. Each theorem includes a short proof sketch, an explanation of the mechanism, and four worked examples. The methods are valid precisely because 2 and 5 divide the base 10, allowing the last digit to fully determine the remainder modulo of these divisors. The work is intended as an accessible piece of recreational number theory for educational purposes.

We observe that some self-similar measures defined by finite or infinite iterated function systems with overlaps satisfy certain "bounded measure type condition", which allows us to extract useful measure-theoretic properties of iterates of the measure. We develop a technique to obtain a closed formula for the spectral dimension of the Laplacian defined by self-similar measure satisfying this condition. For Laplacians defined by fractal measures with overlaps, spectral dimension has been obtained earlier only for a small class of one-dimensional self-similar measures satisfying Strichartz second-order self-similar identities. The main technique we use relies on the vector-valued renewal theorem proved by Lau, Wang and Chu.

In this paper, we will prove the Weyl's law for the asymptotic formula of Dirichlet eigenvalues on metric measure spaces with generalized Ricci curvature bounded from below.

We describe an inductive machinery to prove various properties of representations of a category equipped with a generic shift functor. Specifically, we show that if a property (P) of representations of the category behaves well under the generic shift functor, then all finitely generated representations of the category have the property (P). In this way, we obtain simple criteria for properties such as Noetherianity, finiteness of Castelnuovo-Mumford regularity, and polynomial growth of dimension to hold. This gives a systemetic and uniform proof of such properties for representations of the categories $\FI_G$ and $\OI_G$ which appear in representation stability theory.