, 2020

(a) Let $p$ and $q$ be different prime numbers, and let $a = p^2$, $b = pq$ and $c = p^3q$. Which of $ab$, $bc$ and $ca$ is a square number?

Halle would like to place nine of the ten integers $1, 2, ..., 9, 10$ into the cells of a $3 \times 3$ grid in such a way that the three numbers within each row are written in increasing order from left to right and the product of the three numbers within each row is equal to a square number.

(b) Give an example showing that Halle's task is possible.

(c) How many different grids can Halle produce?