The cardinality of some symmetric differences

In this paper, we prove that for positive integers $k$ and $n$, the cardinality of the symmetric differences of $\{1,2,\dots,k\}$, $\{2,4,\dots,2k\}$, $\{3,6,\dots,3k\}$, ..., $\{n,2n,\dots,kn\}$ is at least $k$ or $n,$ whichever is larger. This solved a problem raised by Pilz in which binary composition codes were studied.