20. Donggyu wants to complete the diagram so that each box contains a positive integer and each box in the top two rows contains the product of the integers in the two boxes below it. He wants the integer in the top box to be 720 . How many different values can the integer $n$ take?

A 1
B 4
C 5
D 6
E 8
SolUtion
D

Let the integers in the left- and right-hand box of the bottom row be $x$ and $y$ respectively, as shown. Since each box in the second and third rows contain the product of the integers in the two boxes below them, the integers in the second row are $x n$ and $y n$ and the integer in the top row is $x y n^{2}$. The question tells us that this is to be equal to 720 and therefore $n$ can be any integer such that its square is a factor of 720 .
When 720 is written as a product of its prime factors, we obtain $720=2^{4} \times 3^{2} \times 5$. Therefore the possible values of $n$ are $1,2,2^{2}, 3,2 \times 3$ and $2^{2} \times 3$. Hence there are six possible values of $n$.