In a product pyramid, each field contains a positive integer that is the product of the numbers in the two adjacent fields from the lower level. At the top of a three-level product pyramid, the number is 90.

![](https://cdn.mathpix.com/cropped/2024_04_17_a7ae5c887c970a6f6dc9g-1.jpg?height=254&width=485&top_left_y=1615&top_left_x=791)

What number can be in the marked field? Determine all possibilities.

(A. Bohiniková)
Let the unknown numbers in the bottom layer of the pyramid be denoted as $a, b, c$. In the second layer, the numbers will be $a b, b c$, and in the top layer, the number will be $a b^{2} c$.

The task is to find the number $b$ such that $a b^{2} c = 90$, i.e., to find the square of an integer in the factorization of the number 90. The prime factorization of the number 90 is

$$
90 = 2 \cdot 3^{2} \cdot 5 \text{.}
$$

Thus, the number in the marked field can be either 1 or 3.

Evaluation. 2 points for the consideration describing the number at the top of the pyramid in the form $a b^{2} c$; 2 points for the factorization of the number $90$; 2 points for all solutions.