1. In a $4 \times 4$ table, 16 different natural numbers are arranged. For each row and each column of the table, the greatest common divisor (GCD) of the numbers located in it was found. It turned out that all eight found numbers are different. For what largest $n$ can we assert that there is a number in such a table that is not less than $n ?$

(A. Khryabrov)
# Answer: 32.

Solution. If the greatest common divisor (GCD) in some row is $n$, then there are four numbers in that row that are divisible by $n$, meaning there is a number no less than $4n$. Since the greatest common divisors in all rows are distinct, one of them is certainly no less than 8. Then, in the corresponding row, there must be a number no less than 32. Now, let's provide an example of a table where all numbers are no greater than 32. The greatest common divisors of the rows are $5, 6, 7$, and 8, while those of the columns are 1, 2, 3, and 4.

| 5 | 10 | 15 | 20 |
| :---: | :---: | :---: | :---: |
| 30 | 6 | 18 | 12 |
| 7 | 14 | 21 | 28 |
| 8 | 16 | 24 | 32 |

Remark. The greatest common divisors must certainly be numbers from 1 to 8, and the rows with GCDs of 6, 7, and 8 should be composed of the numbers that appear in the corresponding rows in the example table (possibly in a different order).