In the diagram, a positive integer is hidden behind each star. The integer shown on each star is the product of the integers hidden behind the other three stars. What is the product of all four hidden integers?
![](https://cdn.mathpix.com/cropped/2024_04_17_2efeb2001664c4e9869eg-1.jpg?height=346&width=1526&top_left_y=1946&top_left_x=359)
Suppose that the numbers hidden behind the four stars, in order, are $w, x, y, z$.

Since the number on the first star is 280 , then the product of the numbers behind the other three stars is 280 , or $x y z=280$.

Similarly, the numbers on the other three stars give the equations $w y z=168, w x z=105$, and $w x y=120$.

Multiplying these four equations together, we obtain

$$
(x y z)(w y z)(w x z)(w x y)=(280)(168)(105)(120)
$$

or

$$
w^{3} x^{3} y^{3} z^{3}=\left(2^{3} \cdot 5 \cdot 7\right)\left(2^{3} \cdot 3 \cdot 7\right)(3 \cdot 5 \cdot 7)\left(2^{3} \cdot 3 \cdot 5\right)
$$

and so

$$
(w x y z)^{3}=2^{9} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3}
$$

Therefore, the product of the four hidden integers, or $w x y z$, equals $2^{3} \cdot 3 \cdot 5 \cdot 7=840$.