Vasiniev N. 5.

Can the numbers be arranged in a $4 \times 4$ table such that the following conditions are simultaneously satisfied:

1) the products of the numbers in each row are the same for all rows;
2) the products of the numbers in each column are the same for all columns;
3) there are no equal numbers among them;
4) all numbers are no greater than 100?
Consider a Graeco-Latin square (the table on the left). In it, each letter appears exactly once in each row and each column, and all 64 pairs of letters in the cells are different.

| $A a$ | $B b$ | $C c$ | $D d$ |
| :--- | :--- | :--- | :--- |
| $B d$ | $A c$ | $D b$ | $C a$ |
| $C b$ | $D a$ | $A d$ | $B c$ |
| $D c$ | $C d$ | $B a$ | $A b$ |


| 1 | 14 | 33 | 65 |
| :--- | :--- | :--- | :--- |
| 26 | 11 | 35 | 3 |
| 21 | 5 | 13 | 22 |
| 55 | 39 | 2 | 7 |

By substituting these eight letters with mutually prime numbers and calculating the corresponding products in the cells, we obtain a table that satisfies the first three conditions. And if $A=a=1, B=2, C=3, D=5 ; b$ $=7, c=11, d=13$, then the fourth condition is also satisfied (see the table on the right).

## Answer

Yes.