The differential cross sections for inclusive neutral pions as a function of transverse and longitudinal momentum in the very forward-rapidity region have been measured at the LHC with the LHC forward detector in proton-proton collisions at $s=2.76$ and $7\,$TeV and in proton-lead collisions at nucleon-nucleon center-of-mass energies of $s_{\rm NN}=5.02\,$TeV. Such differential cross sections in proton-proton collisions are compatible with the hypotheses of limiting fragmentation and Feynman scaling. Comparing proton-proton with proton-lead collisions, we find a sizable suppression of the production of neutral pions in the differential cross sections after subtraction of ultraperipheral proton-lead collisions. This suppression corresponds to the nuclear modification factor value of about $0.1-0.3$. The experimental measurements presented in this paper provide a benchmark for the hadronic interaction Monte Carlo simulation codes that are used for the simulation of cosmic ray air showers.
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