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:

To check for divisibility by 16, you can check for divisibility by 8 and then by 2. Or, more simply, check if the number is divisible by 2 four times. For 720: \(720/2=360\), \(360/2=180\), \(180/2=90\), \(90/2=45\). Since it's divisible by 2 four times, it's divisible by 16.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We need to use the formula for the sum of the interior angles of a polygon and test values based on the given conditions.
Step 2: Key Formula or Approach:
The sum of the interior angles of a polygon with \(n\) sides is given by the formula: \[ \text{Sum of angles} = (n - 2) \times 180^\circ \] We are given two conditions: 1. \(n<10\) 2. The sum of the angles is divisible by 16.
Step 3: Detailed Explanation:
We will test the possible values of \(n\) (number of sides) starting from a triangle (\(n=3\)) up to \(n=9\), as \(n<10\).


n = 3 (Triangle): Sum = \((3-2) \times 180 = 180\). Is 180 divisible by 16? \(180 \div 16 = 11.25\). No.
n = 4 (Quadrilateral): Sum = \((4-2) \times 180 = 360\). Is 360 divisible by 16? \(360 \div 16 = 22.5\). No.
n = 5 (Pentagon): Sum = \((5-2) \times 180 = 540\). Is 540 divisible by 16? \(540 \div 16 = 33.75\). No.
n = 6 (Hexagon): Sum = \((6-2) \times 180 = 720\). Is 720 divisible by 16? \(720 \div 16 = 45\). Yes.
n = 7 (Heptagon): Sum = \((7-2) \times 180 = 900\). Is 900 divisible by 16? \(900 \div 16 = 56.25\). No.
n = 8 (Octagon): Sum = \((8-2) \times 180 = 1080\). Is 1080 divisible by 16? \(1080 \div 16 = 67.5\). No.
n = 9 (Nonagon): Sum = \((9-2) \times 180 = 1260\). Is 1260 divisible by 16? \(1260 \div 16 = 78.75\). No.
The only number of sides less than 10 for which the sum of the interior angles is divisible by 16 is 6.
Step 4: Final Answer:
Polygon A has 6 sides. This corresponds to option (C).