11. In the number triangle shown, each disc is to be filled with a positive integer. Each disc in the top or middle row contains the number which is the product of the two numbers immediately below.
What is the value of $n$ ?
A 1
B 2
C 3
D 6
E 33
SolUTion
A
We let $x$ and $y$ be the positive integers in the discs in the middle row, as shown.

Since these are the products of the two numbers immediately below, $n$ is a factor of both $x$ and $y$. Hence $n^{2}$ is a factor of $x y$.
Now $x y=2022$. It follows that $n^{2}$ is a factor of 2022 .

The prime factorization of 2022 is
$$
2022=2 \times 3 \times 337 \text {. }
$$

It follows that $1^{2}$ is the only square of a positive integer that is a factor of 2022 . We deduce that $n=1$.

FOR INVESTIGATION
11.1 Check that $2 \times 3 \times 337$ is the factorization of 2022 into prime factors.
11.2 In the number triangle shown on the right, each disc is to be filled with a positive integer. Each disc in the top or middle row contains the number which is the product of the two numbers immediately below.
What are the possible values of $n$ ?