Prime factor question and geometry

Question

I have a shape that is less than 9 sides. The sum of its interior angles is divisible by 16. How many sides does this shape have?

So, here's how I started thinking about this problem. The formula for interior angles is 180(n-2) where is the number of sides. So, If the shape is a triangle, then the sum of its interior angles is 180(1) = 180. If the shape is a square, then the sum of its interior angles is 180(2) = 360. If the shape has 5 sides, then the sum of its interior angles is 180(3) = 540.

So the numbers are multiples of 180.

So, which is these numbers is divisible by 16? I'm trying to think of a fast way to do this.

180's prime factors are let's see... : 9 * 20 = 3 * 3 * 10 * 2 = 3 * 3 * 5 * 2 * 2 16's prime factors are : 2 * 2 * 2 * 2

Is there a fast way for me to do this with prime factor and divisor rules? I seem to have forgotten how to go from here.

Answer 1

You have most of the steps already, let me just write them out (again).

Like you mentioned, if the figure has $n$ sides, then the sum of interior angles is $(n-2) 180$.

We are given that $16 \mid (n-2) 180$.

Hint: This implies that $4 \mid (n-2)$.
Hence, what are the possible values of $n$?

Answer 2

Hmmm... I'm trying to comprehend Calvin's hint. I don't think the hint is registering with me because I seem to have forgotten rules about prime factorization.

So the question is just this: 180(n-2) is divisible by 16 and n < 9. What is n?

Again the prime factors of 180 are = 90 * 2 = 10 * 9 * 2 = 2 * 3 * 3 * 5 * 2 And the prime factors of 16 are = 8 * 2 = 4 * 2 * 2 = 2 * 2 * 2 * 2.

I remember this being the first step in figuring out if a number is divisible by another, but I forget how to proceed. and why?