The angles of a polygon are in arithmetic progression. If the smallest angle is 120 degrees and the common difference is 10 degrees, then how many sides does the polygon have ?

The least angle is 120 degrees.
The minimum number of sides required to form a polygon is 3.
If the required polygon is of 3 sides, then the angles of the polygon will be 120, 130 and 140 as they must be in AP with common difference 10 degrees.
Sum of angles = 390 degrees.
But, a polygon of 3 sides must have sum of its angles as 180 degrees.

If the required polygon is of 4 sides, then the angles will be 120, 130, 140 and 150 degrees.
Sum of angles = 540 degrees.
But, a polygon of 4 sides must have sum of its angles 360 degrees.

If the required polygon is of 5 sides, then the angles will be 120, 130, 140, 150 and 160 degrees.
Sum of angles = 700 degrees.
But, a polygon of 5 sides must have sum of its angles 540 degrees.

If the required polygon is of 6 sides, then the angles will be 120, 130, 140, 150, 160 and 170 degrees.
Sum of angles = 870 degrees.
But, a polygon of 6 sides must have sum of its angles 720 degrees.

The polygon cannot be of 7 sides because that will mean one angle must be 180 degrees which is not possible.

Hence, there is no such polygon with the required constraints.

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