We give conditions under which the number of solutions of a system of polynomial equations over a finite field Fq of characteristic p is divisible by p. Our setup involves the substitution ti → f(ti) for auxiliary polynomials f1, . . ., fn ∈ Fq[t]. We recover as special cases results of Chevalley-Warning and Morlaye-Joly. Then we investigate higher p-adic divisibilities, proving a result that recovers the Ax-Katz theorem. We also consider p-weight degrees, recovering work of Moreno-Moreno, Moreno-Castro, and Castro-Castro-Velez.