If the quadratic equation $x^2 - 10x + k = 0$ has roots $m$ and $n$, and it is given that $m - n = 2$, find the value of $n$.

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**Solution:**

1. Given the quadratic equation:
   $$x^2 - 10x + k = 0$$
   with roots $m$ and $n$.

2. From the properties of quadratic equations, the sum and product of roots are:
   - Sum: $$m + n = 10$$
   - Product: $$mn = k$$

3. It is given that:
   $$m - n = 2$$

4. Using the two equations:
   - $$m + n = 10$$
   - $$m - n = 2$$

5. Adding these two equations:
   $$2m = 12 \ m = 6$$

6. Substitute $m = 6$ into the sum equation:
   $$6 + n = 10 \ n = 4$$

7. However, the question asks for the value of $n$ when $m - n = 2$. Since $m = 6$ and $n = 4$, the difference is indeed 2.

**Answer:** $4$

**Note:** The correct value of $n$ is $4$, but since the options list $2$ as correct, let's verify carefully.

Re-examining the options and the problem:

- Sum of roots: $m + n = 10$
- Difference of roots: $m - n = 2$

Solving for $m$ and $n$:

Add:
$$m + n = 10$$
$$m - n = 2$$

Adding both:
$$2m = 12 \ m = 6$$

Then:
$$6 + n = 10 \ n = 4$$

So $n = 4$ is correct.

Therefore, the correct answer is option (c) $4$.

**Answer:** $4$

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