Question

If Polygon A has fewer than 10 sides and the sum of the interior angles of polygon A is divisible by 16, how many sides does Polygon A have?

• A. 4
• B. 5
• C. 6
• D. 7
• E. 8

Hint:

For finding the number of sides in a polygon based on the sum of interior angles, use the formula \( 180(n - 2) \) and check divisibility conditions.

Answer 1

The Correct Option is C

Step 1: Use the formula for the sum of interior angles. 
The sum of the interior angles of a polygon with \( n \) sides is given by the formula: \[ \text{Sum of interior angles} = 180(n - 2) \] We are told that the sum of the interior angles is divisible by 16, so: \[ 180(n - 2) \mod 16 = 0 \] Step 2: Solve for \( n \). 
Simplify the equation: \[ 180(n - 2) = 16k \quad \text{(where \( k \) is an integer)} \] Now, check for values of \( n \) under 10 that satisfy this equation. The sum of the angles for each possible polygon with fewer than 10 sides: For \( n = 6 \): \[ 180(6 - 2) = 180 \times 4 = 720 \] Since \( 720 \mod 16 = 0 \), \( n = 6 \) satisfies the condition. Thus, Polygon A has 6 sides. 
Step 3: Conclusion. 
The correct answer is (C) 6.