Problem

Source: 2016 Saudi Arabia Pre-TST Level 4+ 1.4

Tags: combinatorics, table



The natural numbers $0, 1, 2, 3, . . .$ are written on the square table $2015\times 2015$ in a circular order (anti-clockwise) such that $0$ is in the center of the table. The rows and columns are labelled from bottom to top and from left to right respectively. (see figure below) 1. The number $2015$ is in which row and which column? 2. We are allowed to perform the following operations: First, we replace the number $0$ in the center by $14$, after that, each time, we can add $1$ to each of $12$ numbers on $12$ consecutive unit squares in a row, or $12$ consecutive unit squares in a column, or $12$ unit squares in a rectangle $3\times 4$. After a finite number of steps, can we make all numbers on the table are multiples of $2016$?