We provide a new division formula for holomorphic mappings. It is given in terms of residue currents and has the advantage of being more explicit and simpler to prove than the previously known formulas. 1. Let X be a connected complex manifold with dimC X = n, and f : X → C a holomorphic map. Given also a holomorphic function h on X, we consider the problem of determining whether or not h is “divisible” by f in the sense that h belongs to If , the ideal (in some ring of holomorphic functions on X) generated by f1, . . . , fp. And in case this holds we wish to find a “quotient”, that is, a new holomorphic map g : X → C such that h = g · f = ∑ gjfj . These two questions lie at the heart of the so-called fundamental principle for systems of linear partial differential equations with constant coefficients (see for instance [3] and [1]). And the more explicit the solution of the division problem, the more explicit the fundamental principle. We shall restrict our attention to the case where f is a complete intersection, that is, dimC f−1(0) = n − p, and we shall prove a representation formula of the following type: