Vasiniev N. 5.

Can natural numbers be placed in a $9 \times 9$ 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) no two numbers are equal;
4) all numbers are no greater than 1991?
As in problem 98107, we will construct a $9 \times 9$ Graeco-Latin square (see figure). In it:

1) each cell contains one Latin and one Greek letter;
2) in each row and each column, all letters are different;
3) in any two different cells, the pairs of letters are different.

|  | $b$ |  | $d \delta$ |  |  | $\eta$ |  |  |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
|  | $c \alpha$ |  |  |  |  | $h \zeta$ |  | \| |
|  | $d \ell$ |  |  |  |  | $\varepsilon$ |  | b) |
|  |  |  |  |  |  | $\delta$ |  |  |
|  | $f \eta$ |  |  |  |  |  |  |  |
|  | $g \zeta$ |  |  |  |  |  | $\underline{\gamma}$ |  |
|  | $h \varepsilon$ |  | $a \eta$ | $b \theta$ | cl | $d \alpha$ | $\beta$ | 1 |
|  | $i \delta$ |  | $b \zeta$ | $\sqrt{17}$ | $d \theta$ | $e_{\ell}$ | $\alpha$ | 01 |
|  | 1 |  | $\left.\right\|_{\varepsilon}$ |  | en | $f \theta$ |  |  |

We will assign eighteen pairwise distinct and pairwise coprime numbers to the selected letters and write in the table instead of the pairs of letters the product of the corresponding numbers. The product of all numbers in each column and each row will be equal to the product of all eighteen numbers. If, for example, $a = 1, b = 2, c = 3, d = 5, e = 7, f = 11, g = 13, h = 17, i = 19, \alpha = 23, \beta = 29, \gamma = 31, \delta = 37, \varepsilon = 41, \zeta = 43, \eta = 47, \theta = 53, \imath = 59$, then all numbers in the table will be different, and the maximum number $i \iota = 19 \cdot 59 = 1121$.

## Answer

Yes, it is possible.