MathDB

Let

n \geq 2

be a given integer. At any point

(i, j)

with

i, j \in\mathbb{ Z}

we write the remainder of

i+j

modulo

n

. Find all pairs

(a, b)

of positive integers such that the rectangle with vertices

(0, 0)

(a, 0)

(a, b)

(0, b)

has the following properties: (i) the remainders

0, 1, \ldots , n-1

written at its interior points appear the same number of times; (ii) the remainders

0, 1, \ldots , n -1

written at its boundary points appear the same number of times.

Problem Statement