BALTIC-WAY 2023 Problem 20

Let $n$ be a positive integer. A German set in an $n \times n$ square grid is a set of $n$ cells which contains exactly one cell in each row and column. Given a labelling of the cells with the integers from 1 to $n^{2}$ using each integer exactly once, we say that an integer is a German product if it is the product of the labels of the cells in a German set.

(a) Let $n=8$. Determine whether there exists a labelling of an $8 \times 8$ grid such that the following condition is fulfilled: The difference of any two German products is always divisible by 65 .

(b) Let $n=10$. Determine whether there exists a labelling of a $10 \times 10$ grid such that the following condition is fulfilled: The difference of any two German products is always divisible by 101.