If `A={n:(n^(3)+5n^(2)+2)/(n)` is an integer and n itself is an integer}, then the number of elements in the set A is

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem, we need to determine the set \( A = \{ n : \frac{n^3 + 5n^2 + 2}{n} \text{ is an integer and } n \text{ is an integer} \} \).

### Step-by-Step Solution:

1. **Simplify the Expression**:
We start with the expression \( \frac{n^3 + 5n^2 + 2}{n} \).
\[ \frac{n^3 + 5n^2 + 2}{n} = n^2 + 5n + \frac{2}{n} \]
For this expression to be an integer, \( \frac{2}{n} \) must also be an integer.

2. **Determine Conditions for \( \frac{2}{n} \)**:
The term \( \frac{2}{n} \) is an integer if \( n \) is a divisor of 2. The divisors of 2 are:
\[ n = \pm 1, \pm 2 \]

3. **List Possible Values of \( n \)**:
From the divisors of 2, we find the possible integer values for \( n \):
- \( n = 1 \)
- \( n = -1 \)
- \( n = 2 \)
- \( n = -2 \)

4. **Count the Elements in Set \( A \)**:
The possible integer values of \( n \) that satisfy the condition are \( 1, -1, 2, -2 \). Thus, the number of elements in set \( A \) is:
\[ |A| = 4 \]

### Final Answer:
The number of elements in the set \( A \) is \( 4 \).