Two strings at Hamming distance 1 cannot be both quasiperiodic

We present a generalization to quasiperiodicity of a known fact from combinatorics on words related to periodicity. A string is called periodic if it has a period which is at most half of its length. A string w is called quasiperiodic if it has a non-trivial cover, that is, there exists a string c that is shorter than w and such that every position in w is inside one of the occurrences of c in w. It is a folklore fact that two strings that differ at exactly one position cannot be both periodic. Here we prove a more general fact that two strings that differ at exactly one position cannot be both quasiperiodic. Along the way we obtain new insights into combinatorics of quasiperiodicities.