Olson's theorem

Theorem (Olson ). Let q be a prime power, and let A be an $$m \times n$$ integer matrix, where $$m \geq (q-1)n+1$$ . There exists a non-empty subset of the columns of A such that every coordinate of the sum of these columns is divisible by q.

This is actually a special case of a theorem of Olson on finite Abelian groups that appeared in .