SOLUTION: The exterior angles of polygon are 31 and 26 ; interior angles are 154 each and the remaining interior angles are 171 each. How many sides does the polygon have?

Interior angles are supplementary to exterior angles.  Since it has
some exterior angles of 31°, it has some interior angles of 180°-31° or
149° each.  So there are several possibilities.  For example it could be:

a polygon of 14 sides with 

 6 154° interior angles and  6 26° exterior angles  
 2 171° interior angles and  2  9° interior angles
 6 149° interior angles and  6 31° interior angles
-------------------------------------------------
Sum of interior angles = (n-2)×180° = (14-2)×180° = 12×180° = 2160° 
and 6×154°+2×171°+6×149° = 924°+342°+894° = 2160°. That checks. Also
Sum of exterior angles = 6×26°+2×9°+6×31°=156°+18°+186° = 360°, which
also checks.

Or it could be a polygon of 17 sides with

 7 154° interior angles and  7 26° exterior angles  
 6 171° interior angles and  6  9° interior angles
 4 149° interior angles and  4 31° interior angles
-------------------------------------------------
Sum of interior angles = (n-2)×180° = (17-2)×180° = 15×180° = 2700° 
and 7×154°+6×171°+4×149° = 1078°+1026°+596° = 2700°. That checks. Also
Sum of exterior angles = 7×26°+6×9°+4×31°=182°+54°+124° = 360°, which
also checks.

Or it could be a polygon of 24 sides with

 2 154° interior angles and   2 26° exterior angles  
17 171° interior angles and  17  9° interior angles
 5 149° interior angles and   5 31° interior angles
-------------------------------------------------
Sum of interior angles = (n-2)×180° = (24-2)×180° = 22×180° = 3960° 
and 2×154°+17×171°+5×149° = 308°+2907°+745° = 3960°. That checks. Also
Sum of exterior angles = 2×26°+17×9°+5×31°=52°+153°+155° = 360°, which
also checks.

Or it could be a polygon of 30 sides with

 4 154° interior angles and   4 26° exterior angles  
25 171° interior angles and  25  9° interior angles
 1 149° interior angle  and   1 31° interior angle
-------------------------------------------------
Sum of interior angles = (n-2)×180° = (30-2)×180° = 28×180° = 5040° 
and 4×154°+25×171°+1×149° = 616°+4275°+149° = 5040°. That checks. Also
Sum of exterior angles = 4×26°+25×9°+1×31°=104°+225°+31° = 360°, which
also checks.

So either you've left out some information, or else there are about 6 
or 7 different solutions.  You may discuss it further in the thank-you
note form, and I'll get back to you.

Edwin