If p is a prime and both roots of `x^(2)-px-444p=0` are integers, then p is equal to

Options

A. 2
B. 3
C. 37
D. none of these

Answer

To solve the problem, we need to determine the value of the prime number \( p \) such that both roots of the quadratic equation \( x^2 - px - 444p = 0 \) are integers.

### Step-by-Step Solution

1. **Identify the Quadratic Equation**:
The given quadratic equation is:
\[ x^2 - px - 444p = 0 \]

2. **Calculate the Discriminant**:
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ D = b^2 - 4ac \]
Here, \( a = 1 \), \( b = -p \), and \( c = -444p \). Thus,
\[ D = (-p)^2 - 4 \cdot 1 \cdot (-444p) = p^2 + 1776p \]

3. **Set the Discriminant as a Perfect Square**:
For the roots to be integers, the discriminant must be a perfect square:
\[ D = p^2 + 1776p = k^2 \quad \text{for some integer } k \]
Rearranging gives:
\[ p^2 + 1776p - k^2 = 0 \]

4. **Factor the Quadratic**:
The discriminant \( D \) must be a perfect square, so we can express it as:
\[ D = p(p + 1776) \]
We need \( p(p + 1776) \) to be a perfect square.

5. **Analyze the Expression**:
Since \( p \) is a prime number, we will test small prime numbers to see if \( p(p + 1776) \) is a perfect square.

6. **Testing Values of \( p \)**:
- For \( p = 2 \):
\[ D = 2(2 + 1776) = 2 \times 1778 = 3556 \quad \text{(not a perfect square)} \]
- For \( p = 3 \):
\[ D = 3(3 + 1776) = 3 \times 1779 = 5337 \quad \text{(not a perfect square)} \]
- For \( p = 5 \):
\[ D = 5(5 + 1776) = 5 \times 1781 = 8905 \quad \text{(not a perfect square)} \]
- For \( p = 7 \):
\[ D = 7(7 + 1776) = 7 \times 1783 = 12481 \quad \text{(not a perfect square)} \]
- For \( p = 37 \):
\[ D = 37(37 + 1776) = 37 \times 1813 = 67181 \]
We check if \( 67181 \) is a perfect square:
\[ \sqrt{67181} = 259 \quad \text{(which is an integer)} \]

7. **Conclusion**:
Since \( D \) is a perfect square when \( p = 37 \) and \( p \) is a prime number, we conclude:
\[ p = 37 \]

### Final Answer:
The value of \( p \) is \( \boxed{37} \).