Motivated by recent experiments on small radius nanotubes, we study the superconducting instabilities of cylindrical (5,0) nanotubes. According to band structure calculations, thesenanotubes 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 bands model we consider for the description of these (5,0) cylindrical nanotubes.