Motivated by recent experiments on small radius nanotubes, we study the superconducting instabilities of cylindrical (5,0) nanotubes. According to band-structure calculations, these nanotubes possess three bands at the Fermi energy. Using a Fermionic renormalization-group approach and a careful Bosonization treatment, we consider the effect of different attractive interactions, mediated by phonons, within the Luttinger liquid framework. We particularly focus on a superconducting instability specific to the three-band model we consider for the description of these (5,0) cylindrical nanotubes.