I'm not an expert in this area, but I've heard that algebraic topologists run into congruences quite often. For example, the stable homotopy groups of spheres are almost always finite abelian groups, and are often studied by examining the Sylow subgroups with $p$-specific toolsets.

On the subject of infinite series, one encounters congruences working with the spectrum $tmf$ of topological modular forms. It admits a map from $\pi_0$ to the ring of level 1 modular forms with integer coefficients that is an isomorphism after 6 is inverted. The image satisfies some congruences mod powers of 2 and 3 that are akin to the Borcherds congruences satisfied by theta functions. For example, the cusp form $\Delta$ does not appear in the image, but $24\Delta$ and $\Delta^{24}$ do appear.

I'm not an expert in this area, but I've heard that algebraic topologists run into congruences quite often. For example, the stable homotopy groups of spheres are almost always finite abelian groups, and are often studied by examining the Sylow subgroups with $p$-specific toolsets.

On the subject of infinite series, one encounters congruences working with the spectrum $tmf$ of topological modular forms. It admits a map from $\pi_0$ to the ring of level 1 modular forms with integer coefficients that is an isomorphism after 6 is inverted. The image satisfies some congruences mod powers of 2 and 3 that are akin to the Borcherds congruences satisfied by theta functions. For example, the cusp form $\Delta$ does not appear in the image, but $24\Delta$ and $\Delta^{24}$ do appear.