sums of discrete uniform random variables

Let X_1, X_2, X_3, ... be independent, identically distributed random
variables, each uniformly distributed on the set {0,..., n-1}. In
other words, each X_i is uniformly distributed mod n.

Prove or disprove:

(X_1 + ... + X_k) mod m is uniformly distributed mod m iff m|n.

quasi
.