# Olson's theorem

From Egres Open

**Theorem (Olson ^{[1]}).** Let

*q*be a prime power, and let

*A*be an [math]m \times n[/math] integer matrix, where [math]m \geq (q-1)n+1[/math]. 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 ^{[1]}.

## References

- ↑
^{1.0}^{1.1}J.E. Olson,*A combinatorial problem on finite Abelian groups I*, J. Number Theory 1 (1969) 8-10, DOI link