A positive integer is to be placed in each box. The product of any four adjacent integers is always 120. What is the value of $x$ ?

|  |  | 2 |  |  | 4 |  |  | $x$ |  |  | 3 |  |  |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
A positive integer is to be placed in each box. The product of any four adjacent integers is always 120. What is the value of $x$ ?

|  |  | 2 |  |  | 4 |  | $x$ |  |  |  | 3 |  |  |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

## Solution

Since the product of any four integers is $120, a_{1} a_{2} a_{3} a_{4}=a_{2} a_{3} a_{4} a_{5}=120$ where $a_{n}$ represents the number in the $n$th box. Therefore, $a_{1}=a_{5}$ and similarly $a_{2}=a_{6}, a_{3}=a_{7}, a_{4}=a_{8}$ or more generally, $a_{n}=a_{n+4}$. Thus the boxes can be filled as follows:

| $x$ | 4 | 2 | 3 | $x$ | 4 | 2 | 3 | $x$ | 4 | 2 | 3 | $x$ | 4 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |

Therefore, $(4)(2)(3)(x)=120$

$$
x=\frac{120}{24}=5
$$

ANSWER: (E)