What is the smallest possible natural number 'n' for which the equation `x^(2)-nx + 2014=0` has integer roots.

Answer

To find the smallest possible natural number \( n \) for which the equation

\[ x^2 - nx + 2014 = 0 \]

has integer roots, we can follow these steps:

### Step 1: Understand the relationship between roots and coefficients
For a quadratic equation of the form \( x^2 - nx + c = 0 \), if \( \alpha \) and \( \beta \) are the roots, then by Vieta's formulas, we have:

\[ \alpha + \beta = n \quad \text{(1)} \]
\[ \alpha \beta = c \quad \text{(2)} \]

In our case, \( c = 2014 \). Therefore, we have:

\[ \alpha \beta = 2014 \]

### Step 2: Factor 2014
To find integer roots, we need to factor 2014. We can start by finding its prime factorization:

\[ 2014 = 2 \times 1007 \]

Next, we factor 1007:

\[ 1007 = 19 \times 53 \]

Thus, the complete prime factorization of 2014 is:

\[ 2014 = 2^1 \times 19^1 \times 53^1 \]

### Step 3: Find pairs of factors
Next, we need to find all pairs of integers \( (\alpha, \beta) \) such that \( \alpha \beta = 2014 \). The pairs of factors of 2014 are:

- \( (1, 2014) \)
- \( (2, 1007) \)
- \( (19, 106) \)
- \( (38, 53) \)
- \( (53, 38) \)
- \( (106, 19) \)
- \( (1007, 2) \)
- \( (2014, 1) \)

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

### Step 5: Identify the smallest \( n \)
From the calculated values of \( n \):

- \( 2015, 1009, 125, 91, 91, 125, 1009, 2015 \)

The smallest value is \( 91 \).

### Conclusion
Thus, the smallest possible natural number \( n \) for which the equation \( x^2 - nx + 2014 = 0 \) has integer roots is

\[ \boxed{91} \]