I've been given this as part of my degree and I'm struggling to solve it:

Suppose A is a 10x10 matrix with integer entries, with the following property: given any five rows and five columns, the sum of the entries of the 5x5 matrix formed by these rows and columns is even. Prove that all the entries of A are even.

I've tried proving it first for any 1 row and 5 columns (forming a 1x5 matrix) so that I can then possibly use induction but I'm still struggling.

Any help would be greatly appreciated! Thanks!