Let T be an infinite array in which every entry is 1. This satisfies the divisibility requirement, but not the requirement that the entries are distinct. We will replace the entries of T with distinct values, and after each step we will restore the divisibility.
When T(i, j) is the next entry to be replaced, assume by induction that T satisfies the divisibility requirement. In particular, let c be the (i - 1)th partial sum of the jth column, let r be the (j - 1)th partial sum of the ith row, and let m be the current value of T(i, j). Then c + m is divisible by i, and r + m is divisible by j. We replace m with a new value n, so that c + n is also divisible by i, and r + n is also divisible by j. Then we add m - n to T(i + 1, j) and T(i, j + 1), and add n - m to T(i + 1, j + 1). Then the (i + 1)th partial sums of the columns are unchanged, and so are the (j + 1)th partial sums of the rows. The ith partial sums of the jth and (j + 1)th columns have changed by an amount that is divisible by i, so these sums are still divisible by i. Similarly, the jth partial sums of the ith and (i + 1)th rows are still divisible by j. Therefore all the divisibility requirements are still satisfied. This completes the proof.