Question: The interior angles of a convex polygon are in arithmetic progression. The smallest angle is 120°and the common difference is 5°. Find the number of sides of the polygon if the polygon has more than 10 sides.

1. 15
2. 16
3. 17
4. 18
5. 20

Correct Answer: (B)

Approach Solution 1 :

We know that the sum of interior angles for any polygon is, (n−2)180°.

Therefore the sum of angles will be,

=> (n/2)*[2a+(n−1)d] = (n/2)*(2*120 + (n−1)5)

Thus,

=> (n−2)180° = (n/2)*(2∗120+(n−1)5)
=> 240n + (5n^2) − 5n = 360n − 720
=> (n^2) − 25n + 144 = 0
=> (n^2) − 16n − 9n + 144 = 0
=> (n−16)(n−9)=0

As a result, n can have a value of 16 or 9.

However, the query indicates that the polygon has more than 10 sides, so n=16.

Approach Solution 2 :

The rule for a set of values that are equally spaced apart, the sum of the values = (median) * (Count of Terms)

For any N-sided polygon, the sum of the interior angles = 180 * (N - 2)
where N = a positive integer number of sides that is greater that or equall to 3.
As a result, the following equation must be true no matter what the SUM of the interior angles is,
(Count of Angles/Sides = N) * (Median of Angle Measures) = 180 * (N -2)
=> (Median) * N = 180 * (N - 2)

Due to the fact that the sum of the interior angles will be a positive integer, the sum must be divisible by 180. According to the factor foundation rule, this means that the sum must be divisible by the factors 9, 4, and 5.
Option - E : 20 sides
=> Median = (10th Value + 11th Value)
=> N = 20

SUM = (165 + 170)/2 * 20 = 335 * 10
This sum is not divisible by 9 which results it being not divisible by 180
As a result, this option is eliminated.

Option - D : 18 sides

=> (160 + 165)/2 * 18
=> 325 * 9

This sum is not divisible by 4 which results it being not divisible by 180

As a result, this option is eliminated.

Option - C : 17 sides
=> (160) * 17

This sum is not divisible by 9 which results it being not divisible by 180

As a result, this option is eliminated.

Option - B : 16 side

=> (155 + 160)/2 * 16
=> 315 * 8 —---> The prime factorization = (2)^3 * (3)^2 * (5) * (7)

Prime Factorization of 180 = (2)^2 * (3) * (5)^2
The SUM of the interior Angles of the 16 sides given by this Arithmetic Progression will be a valid sum because the exponents are equal to or greater than the exponents of 180's Prime Factorization.
It can be noted that the first option also will not work in the same way.

“The interior angles of a convex polygon are in arithmetic progression” - is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. GMAT Quant section consists of 31 questions in total. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

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