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(Abridged) We present evidence that low-mass starless cores, the simplest units of star formation, are systematically differentiated in their chemical composition. Molecules including CO and CS almost vanish near the core centers, where the abundance decreases by one or two orders of magnitude. At the same time, N2H+ has a constant abundance, and the fraction of NH3 increases toward the core center. Our conclusions are based on a study of 5 mostly-round starless cores (L1498, L1495, L1400K, L1517B, and L1544), which we have mappedin C18O(1-0), C17O(1-0), CS(2-1), C34S(2-1), N2H+(1-0), NH3(1,1) and (2,2), and the 1.2 mm continuum. For each core we have built a model that fits simultaneously the radial profile of all observed emission and the central spectrum for the molecular lines. The observed abundance drops of CO and CS are naturally explained by the depletion of these molecules onto dust grains at densities of 2-6 10^4 cm-3. N2H+ seems unaffected by this process up to densities of several 10^5, while the NH3 abundance may be enhanced by reactions triggered by the disappearance of CO from the gas phase. With the help of our models, we show that chemical differentiation automatically explains the discrepancy between the sizes of CS and NH3 maps, a problem which has remained unexplained for more than a decade. Our models, in addition, show that a combination of radiative transfer effects can give rise to the previously observed discrepancy in the linewidth of these two tracers. Although this discrepancy has been traditionally interpreted as resulting from a systematic increase of the turbulent linewidth with radius, our models show that it can arise in conditions of constant gas turbulence.