370. In each cell of a $15 \times 15$ square, one of the first 15 natural numbers is written. Two identical numbers cannot be in the same row or column, and in cells symmetric with respect to one of the two main diagonals, the same numbers are written. Can the numbers written on this main diagonal be equal?
$\triangle$ In each row and each column of the square, all natural numbers from 1 to 15 are placed, and each number appears only once. Therefore, each of these numbers appears in the square exactly 15 times.

Take, for example, the number 1. Symmetrically relative to the main diagonal of the square, only an even number of the 15 ones can be placed. Therefore, at least one one is recorded on the diagonal. Since similar reasoning is valid for each of the other natural numbers from 2 to 15, each of the natural numbers from 1 to 15 stands on the diagonal, and exactly once. But then all the numbers on the diagonal are different.

Answer: they cannot.