Let's solve the problem step by step.

Given:

- \( 7056_n - 4577_n = 2257_n \)

We need to find the value of the base \( n \).

First, let's convert the numbers from base \( n \) to base 10:

1. \( 7056_n \) in base 10:
\[
7n^3 + 0n^2 + 5n + 6
\]

2. \( 4577_n \) in base 10:
\[
4n^3 + 5n^2 + 7n + 7
\]

3. \( 2257_n \) in base 10:
\[
2n^3 + 2n^2 + 5n + 7
\]

The equation becomes:

\[
(7n^3 + 0n^2 + 5n + 6) - (4n^3 + 5n^2 + 7n + 7) = 2n^3 + 2n^2 + 5n + 7
\]

Simplifying this:

\[
7n^3 + 5n + 6 - 4n^3 - 5n^2 - 7n - 7 = 2n^3 + 2n^2 + 5n + 7
\]

Combine like terms:

\[
3n^3 - 5n^2 - 2n - 1 = 2n^3 + 2n^2 + 5n + 7
\]

Bring all terms to one side:

\[
3n^3 - 5n^2 - 2n - 1 - 2n^3 - 2n^2 - 5n - 7 = 0
\]

Simplify:

\[
n^3 - 7n^2 - 7n - 8 = 0
\]

To find the value of \( n \), substitute the given options one by one:

1. \( n = 8 \cfrac{17}{30} \):

\[
n = \frac{30 \times 8 + 17}{30} = \frac{240 + 17}{30} = \frac{257}{30}
\]

Check if the base \( n = 8.5667 \):


Answer: A. \( 8 \cfrac{17}{30}\)