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[References.: 53 tit.]

This Letter reports measurements of differential cross sections for the production of two Z bosons in association with jets in proton-proton collisions at s=8 and 13 TeV. The analysis is based on data samples collected at the LHC with the CMS detector, corresponding to integrated luminosities of 19.7 and 35.9fb−1 at 8 and 13 TeV, respectively. The measurements are performed in the leptonic decay modes ZZ→ℓ+ℓ−ℓ′+ℓ′−, where ℓ,ℓ′=e,μ. The differential cross sections as a function of the jet multiplicity, the transverse momentum pT, and pseudorapidity of the pT -leading and subleading jets are presented. In addition, the differential cross sections as a function of variables sensitive to the vector boson scattering, such as the invariant mass of the two pT -leading jets and their pseudorapidity separation, are reported. The results are compared to theoretical predictions and found in good agreement within the theoretical and experimental uncertainties.