maheshsrini wrote:
Can somebody please explain the question and the answer. I am not clear with the question.
BDSunDevil wrote:
for a regular polygon of p sides .The number of values of p= 20 or 24, the polygon will have angles whose values in degrees can be expressed in integers. using the formula 180*(p-2)/p. i. e. only 20 and 24 yield integer value for 180*(p-2)/p.
Can someone please check the answers.
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:
math-polygons-87336.html2.
Finding the Number of Factors of an IntegerFirst 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:
math-number-theory-88376.htmlBACK 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 can not have 1 or 2 sides (p can not 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.
Please enlighten me for that I think there are three 2s in the number of factors of 360. So I would think that 24-1-3=20. Can you clarify me on this?