4. On a magnetic board, there are the first 16 natural numbers not divisible by 4. Elena intends to fill a rectangle with 3 rows and 5 columns with distinct numbers from the board such that the 3 row sums are equal and the 5 column sums are equal. She starts with the largest number on the board and succeeds in her goal.

a) Which number on the board was not used?

b) Provide an example of the arrangement of numbers that satisfies the conditions in the statement!

Gabriel Daniilescu, Brăila
Solution. a) The set of numbers on the board is

$$
A=\{1,2,3,5,6,7,9,10,11,13,14,15,17,18,19,21\}
$$

Their sum is $S_{16}=171$. Let $n$ be the unused number and $S_{15}$ the sum of the numbers placed in the 15 squares of the rectangle. We have $S_{15}=171-n$. Let $S_{c}$ be the sum of the numbers in a column and $S_{l}$ the sum of the numbers in a row $\Rightarrow S_{15}=5 S_{c}$ and $S_{15}=3 S_{l} \Rightarrow S_{15} \vdots 5$ and $S_{15} \vdots 3 \Rightarrow S_{15}: 15 \Rightarrow 171-n: 15 \Leftrightarrow$ $\Leftrightarrow 165+6-n: 15 \Leftrightarrow 6-n: 15 \Leftrightarrow n-6: 15 \Rightarrow n \in\{6,21\}$. But 21 is the largest number on the board $\Rightarrow n \neq 21 \Rightarrow n=6$.
b) $S_{15}=171-6=165 \Rightarrow S_{c}=33$ and $S_{l}=55$. An example would be:

| 5 | 11 | 17 | 13 | 9 |
| :---: | :---: | :---: | :---: | :---: |
| 7 | 19 | 1 | 18 | 10 |
| 21 | 3 | 15 | 2 | 14 |