ILL 1984 Problem 64

For a matrix $(p_{ij})$ of the format $m\times n$ with real entries, set $$a_i = \sum_{j=1}^n p_{ij}\text{ for }i = 1,\cdots,m\text{ and }b_j = \sum_{i=1}^m p_{ij}\text{ for }j = 1, . . . , n\longrightarrow(1)$$ By integering a real number, we mean replacing the number with the integer closest to it. Prove that integering the numbers $a_i, b_j, p_{ij}$ can be done in such a way that $(1)$ still holds.