We develop sufficient conditions for containment of graph subdivisions with subdivided edges of prescribed divisibility in terms of containment of graph minors. Concretely, we prove that for every graph H of maximum degree at most 3 and for every positive integer q there is a finite f=f(H,q) such that every minor of a complete graph K_f contains a subdivision of H in which every edge is replaced by a path whose length is divisible by q. Both the assumption of maximum degree at most 3 and the requirement of zero residue modulo q for path length are essential. For the case of cycles we can do much better - we show that for f=O(q \log q) every K_f-minor contains a cycle of length divisible by q. This result settles a recent problem of Friedman and Krivelevich about cycles in (weakly) expanding graphs.
Joint work with Noga Alon.