The smallest possible natural number n, for which the equation `x^(2) - nx + 2014 = 0` has integral roots, is

Answer

To find the smallest possible natural number \( n \) for which the equation \( x^2 - nx + 2014 = 0 \) has integral roots, we can follow these steps:

### Step 1: Understand the roots of the quadratic equation
For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the sum and product of the roots \( \alpha \) and \( \beta \) can be given by:
- Sum of roots: \( \alpha + \beta = -\frac{b}{a} = n \)
- Product of roots: \( \alpha \beta = \frac{c}{a} = 2014 \)

### Step 2: Set up the equations
From the above, we have:
- \( \alpha + \beta = n \)
- \( \alpha \beta = 2014 \)

### Step 3: Factor 2014
To find integral roots, we need to factor \( 2014 \). The prime factorization of \( 2014 \) is:
\[ 2014 = 2 \times 19 \times 53 \]

### Step 4: Find pairs of factors
We need to find pairs of factors of \( 2014 \) that can serve as \( \alpha \) and \( \beta \). The pairs of factors (both positive) are:
- \( (1, 2014) \)
- \( (2, 1007) \)
- \( (19, 106) \)
- \( (38, 53) \)

### Step 5: Calculate \( n \) for each pair
Now, we calculate \( n \) for each pair:
1. For \( (1, 2014) \):
\[ n = 1 + 2014 = 2015 \]
2. For \( (2, 1007) \):
\[ n = 2 + 1007 = 1009 \]
3. For \( (19, 106) \):
\[ n = 19 + 106 = 125 \]
4. For \( (38, 53) \):
\[ n = 38 + 53 = 91 \]

### Step 6: Identify the smallest \( n \)
From the calculated values of \( n \):
- \( 2015, 1009, 125, 91 \)

The smallest possible natural number \( n \) is:
\[ \boxed{91} \]