I love this topic and look forward to reading the next articles, but I suggest not saying two numbers are "equal" mod N. I would say they are "equivalent" mod N and maybe point out the broader insight: Equality is often too rigid a constraint, and we usually want to consider equivalence relations instead. We know 3 and 6 and 9 are obviously not equal, but it's useful to notice the pattern that they are all divisible evenly by 3, i.e., they are all in the equivalence class 0 mod 3.

When I think about Langlands, I think it is the power of equivalence over equality that shockingly allows us to connect the discrete world of the natural numbers (or Q) with the world of the continuous (R or C), across disparate branches of mathematics. The Modularity Theorem (every elliptic curve over Q is modular) is the foundational idea and at every step along the way, we obtain evidence of more remarkable equivalences: The conductor N of an elliptic curve versus the level N of certain congruence groups; the point count deficiency (p'th Hecke eigenvalue) of a curve and the p'th coefficient of the Fourier q-expansion; Galois reciprocity showing an equivalence between the traces of Frobenius elements acting on a cohomology, and the eigenvalues of Hecke operators; Ribet's theorem about level lowering; etc. Time and again, the theme in Langlands is that equivalence relationships make it possible for us to reason why two intricate mathematical structures that seem completely foreign are actually "essentially the same" -- not equal, but equivalent.