For every integers $ a,b,c$ whose greatest common divisor is $n$, if
\[ \begin{array}{l} {x \plus{} 2y \plus{} 3z \equal{} a} \\
{2x \plus{} y \minus{} 2z \equal{} b} \\
{3x \plus{} y \plus{} 5z \equal{} c} \end{array}
\]
has a solution in integers, what is the smallest possible value of positive number $ n$?

$\textbf{(A)}\ 7   \qquad\textbf{(B)}\ 14   \qquad\textbf{(C)}\ 28   \qquad\textbf{(D)}\ 56   \qquad\textbf{(E)}\ \text{None}$
To solve the problem, we need to determine the smallest possible value of \( n \) such that the system of linear equations has integer solutions for every set of integers \( a, b, \) and \( c \) whose greatest common divisor is \( n \).

Given the system of equations:
\[
\begin{array}{l} 
x + 2y + 3z = a \\ 
2x + y - 2z = b \\ 
3x + y + 5z = c 
\end{array}
\]

We can represent this system in matrix form as:
\[
\begin{pmatrix}
1 & 2 & 3 \\
2 & 1 & -2 \\
3 & 1 & 5 
\end{pmatrix}
\begin{pmatrix}
x \\
y \\
z 
\end{pmatrix}
=
\begin{pmatrix}
a \\
b \\
c 
\end{pmatrix}
\]

Let \( A \) be the coefficient matrix:
\[
A = \begin{pmatrix}
1 & 2 & 3 \\
2 & 1 & -2 \\
3 & 1 & 5 
\end{pmatrix}
\]

We need to find the determinant of \( A \) to determine if the system has a unique solution:
\[
\text{det}(A) = \begin{vmatrix}
1 & 2 & 3 \\
2 & 1 & -2 \\
3 & 1 & 5 
\end{vmatrix}
\]

Calculating the determinant:
\[
\text{det}(A) = 1 \cdot \begin{vmatrix}
1 & -2 \\
1 & 5 
\end{vmatrix}
- 2 \cdot \begin{vmatrix}
2 & -2 \\
3 & 5 
\end{vmatrix}
+ 3 \cdot \begin{vmatrix}
2 & 1 \\
3 & 1 
\end{vmatrix}
\]

\[
= 1 \cdot (1 \cdot 5 - (-2) \cdot 1) - 2 \cdot (2 \cdot 5 - (-2) \cdot 3) + 3 \cdot (2 \cdot 1 - 3 \cdot 1)
\]

\[
= 1 \cdot (5 + 2) - 2 \cdot (10 + 6) + 3 \cdot (2 - 3)
\]

\[
= 1 \cdot 7 - 2 \cdot 16 + 3 \cdot (-1)
\]

\[
= 7 - 32 - 3
\]

\[
= -28
\]

The determinant of \( A \) is \( -28 \). This means that the system has a unique solution if and only if the right-hand side vector \( \begin{pmatrix} a \\ b \\ c \end{pmatrix} \) is divisible by 28. Therefore, the smallest possible value of \( n \) such that the system has integer solutions for every set of integers \( a, b, \) and \( c \) is \( 28 \).

The final answer is \( \boxed{28} \).