Find a positive integer $n$ such that there is a polygon with $n$ sides where each of its interior angles measures $177^o$
1. We start with the formula for the sum of the interior angles of a polygon with \( n \) sides, which is given by:
   \[
   (n-2) \times 180^\circ
   \]
2. Since each interior angle of the polygon measures \( 177^\circ \), the total sum of the interior angles can also be expressed as:
   \[
   177^\circ \times n
   \]
3. Setting these two expressions equal to each other, we get:
   \[
   (n-2) \times 180^\circ = 177^\circ \times n
   \]
4. Expanding and simplifying the equation:
   \[
   180n - 360 = 177n
   \]
5. Isolating \( n \) on one side of the equation:
   \[
   180n - 177n = 360
   \]
   \[
   3n = 360
   \]
6. Solving for \( n \):
   \[
   n = \frac{360}{3} = 120
   \]

The final answer is \( \boxed{120} \)