3A. Given are 21 tiles in the shape of a square, of the same size. On four tiles is written the number 1; on two tiles is written the number 2; on seven tiles is written the number 3; on eight tiles is written the number 4. Using 20 of these tiles, Dimitar formed a rectangle with dimensions 4 by 5. For the formed rectangle, the sum of the numbers in each row is the same, and the sum of the numbers in each column is the same. Which number is on the unused tile?
Solution. Let's denote by $S$ the sum of all numbers written on the tiles that form the rectangle. From the condition of the problem, we have that 4 is a divisor of $S$ and that 5 is a divisor of $S$. This means that 20 is a divisor of $S$. The sum of all numbers written on the 21st tile is exactly 61. We conclude that the number on the unused tile must be 1.