Consider a regular polygon of p sides. The number of values

Couple of things:
1. Sum of interior angles of a polygon is given by \(180(n-2)\) where \(n\) is the number of sides (for example the sum of interior angles of a triangle is \(180(3-2)=180\) degrees and the sum of interior angles of a quadrilateral is \(180(4-2)=360\) degrees).

Question below talks about a regular polygon, which is a polygon with all equal sides and equal interior angles. Thus each interior angle of a regular polygon is given by: \(\frac{180(n-2)}{n}\) (for example each interior angle of an equilateral triangle is \(\frac{180(3-2)}{3}=60\) degrees and each interior angles of a square is \(\frac{180(4-2)}{4}=90\) degrees).
For more on polygons check: https://gmatclub.com/forum/math-polygons-87336.html

2. Finding the Number of Factors of an Integer

First make prime factorization of an integer \(n=a^p*b^q*c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\) and \(p\), \(q\), and \(r\) are their powers.

The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: \(450=2^1*3^2*5^2\)

Total number of factors of 450 including 1 and 450 itself is \((1+1)*(2+1)*(2+1)=2*3*3=18\) factors.
For more on number properties check: https://gmatclub.com/forum/math-number-theory-88376.html

BACK TO THE ORIGINAL QUESTION:
Consider a regular polygon of p sides .The number of values of p for which the polygon will have angles whose values in degrees can be expressed in integers?
A. 24
B. 23
C. 22
D. 20
E. 21

The question is: for how many values of \(p\) (where p is the # of sides of a regular polygon) \(\frac{180(p-2)}{p}\) is an integer (or how many sided regular polygons exist which have interior angles equal to an integer).

Now, \(\frac{180(p-2)}{p}=180-\frac{360}{p}\) to be an integer \(\frac{360}{p}\) must be an integer, so \(p\) must be a factor of 360. How many different positive factors does 360 have? Since \(360=2^3*3^2*5\) then # of factors is \((3+1)(2+1)(1+1)=24\), including 1 and 360. Thus if \(p\) is any of these 24 values (1, 2, 3, ... 360) then \(\frac{180(p-2)}{p}\) is an integer.

Finally, as polygon cannot have 1 or 2 sides (p cannot be 1 or 2) then only 24-2=22 regular polygons exist which have interior angles equal to an integer: 3 sided (equilateral triangle), 4 sided (square), 5 sided (regular pentagon), ..., 360 sided (trictohexacontagon ).

Answer: C.

Hope it's clear.