The measures (in degrees) of the interior angles of a convex hexagon form an arithmetic sequence of positive integers.  Let $m^{\circ}$ be the measure of the largest interior angle of the hexagon.  The largest possible value of $m^{\circ}$ is

$ \textbf{(A)}\ 165^{\circ}\qquad\textbf{(B)}\ 167^{\circ}\qquad\textbf{(C)}\ 170^{\circ}\qquad\textbf{(D)}\ 175^{\circ}\qquad\textbf{(E)}\ 179^{\circ} $
1. **Identify the sum of the interior angles of a hexagon:**
   The sum of the interior angles of a hexagon can be calculated using the formula for the sum of the interior angles of a polygon, which is given by:
   \[
   \text{Sum of interior angles} = (n-2) \times 180^\circ
   \]
   For a hexagon (\(n = 6\)):
   \[
   \text{Sum of interior angles} = (6-2) \times 180^\circ = 4 \times 180^\circ = 720^\circ
   \]

2. **Set up the arithmetic sequence:**
   Let the measures of the interior angles be \(a, a+d, a+2d, a+3d, a+4d, a+5d\), where \(a\) is the smallest angle and \(d\) is the common difference. The largest angle \(m\) is \(a + 5d\).

3. **Sum of the angles in terms of \(a\) and \(d\):**
   The sum of the angles is:
   \[
   a + (a+d) + (a+2d) + (a+3d) + (a+4d) + (a+5d) = 720^\circ
   \]
   Simplifying, we get:
   \[
   6a + 15d = 720^\circ
   \]
   Dividing the entire equation by 3:
   \[
   2a + 5d = 240^\circ
   \]

4. **Express \(a\) in terms of \(d\):**
   Solving for \(a\):
   \[
   2a = 240^\circ - 5d
   \]
   \[
   a = 120^\circ - \frac{5d}{2}
   \]

5. **Determine the largest angle \(m\):**
   The largest angle \(m\) is:
   \[
   m = a + 5d = 120^\circ - \frac{5d}{2} + 5d = 120^\circ + \frac{5d}{2}
   \]

6. **Ensure \(m\) is an integer and less than \(180^\circ\):**
   For \(m\) to be an integer, \(\frac{5d}{2}\) must be an integer, implying \(d\) must be even. Let \(d = 2k\) for some integer \(k\):
   \[
   m = 120^\circ + \frac{5(2k)}{2} = 120^\circ + 5k
   \]
   Since \(m < 180^\circ\):
   \[
   120^\circ + 5k < 180^\circ
   \]
   \[
   5k < 60^\circ
   \]
   \[
   k < 12
   \]
   The largest integer \(k\) satisfying this inequality is \(k = 11\).

7. **Calculate the largest possible value of \(m\):**
   Substituting \(k = 11\) into the expression for \(m\):
   \[
   m = 120^\circ + 5 \times 11 = 120^\circ + 55^\circ = 175^\circ
   \]

The final answer is \(\boxed{175^\circ}\).