  We investigate the one-dimensional $p$-wave superconducting model with
periodically modulated hopping and show that under time-reversal symmetry, the
number of the Majorana zero modes (MZMs) strongly depends on the modulation
period. If the modulation period is odd, there can be at most one MZM. However
if the period is even, the number of the MZMs can be zero, one and two. In
addition, the MZMs will disappear as the chemical potential varies. We derive
the condition for the existence of the MZMs and show that the topological
properties in this model are dramatically different from the one with
periodically modulated potential.
