MathDB

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 thecells with the integers from

1

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 alwaysdivisible 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

20