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# The measures of the interior angles in a polygon are consecutive integ

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The measures of the interior angles in a polygon are consecutive integ [#permalink]
Dear Bunuel VeritasKarishma chetan2u IanStewart MathRevolution,

n = 9 or 80.

However, according to the official solution, it says:
hazelnut wrote:
As it turns out (S-9)(S-80)=0, which means S can be 9 or 80. However S cannot be 80 because this creates a polygon with angles greater than 180.

I think the highlighted part is not right.
If n= 80, the polygon would still be a valid concave polygon since the largest angle would be 136 + (80-1) = 215 degrees right?

Originally posted by kornn on 04 Apr 2020, 02:15.
Last edited by kornn on 04 Apr 2020, 06:26, edited 1 time in total.
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Re: The measures of the interior angles in a polygon are consecutive integ [#permalink]
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rxs0005 wrote:
The measures of the interior angles in a polygon are consecutive integers. The smallest angle measures 136 degrees. How many sides does this polygon have?

A) 8
B) 9
C) 10
D) 11
E) 13

Why use quadratic equation when we can simply look at unit digit and options to answer this question

Check video solution here.

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Screenshot 2020-04-04 at 3.01.43 PM.png [ 913.37 KiB | Viewed 2813 times ]

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Re: The measures of the interior angles in a polygon are consecutive integ [#permalink]
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varotkorn wrote:
Dear Bunuel VeritasKarishma chetan2u IanStewart MathRevolution,

n = 9 or 80.

However, according to the official solution, it says:
hazelnut wrote:
As it turns out (S-9)(S-80)=0, which means S can be 9 or 80. However S cannot be 80 because this creates a polygon with angles greater than 180.

I think the highlighted part is not right.
If n= 80, the polygon would still be a valid concave polygon since the largest angle would be 136 + (80-1) = 215 degrees right?

Whenever we talk of a polygon in GMAT, it is a convex polygon.
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Re: The measures of the interior angles in a polygon are consecutive integ [#permalink]
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varotkorn wrote:
Dear Bunuel VeritasKarishma chetan2u IanStewart MathRevolution,

n = 9 or 80.

However, according to the official solution, it says:
hazelnut wrote:
As it turns out (S-9)(S-80)=0, which means S can be 9 or 80. However S cannot be 80 because this creates a polygon with angles greater than 180.

I think the highlighted part is not right.
If n= 80, the polygon would still be a valid concave polygon since the largest angle would be 136 + (80-1) = 215 degrees right?

Their solution seems to miss the point. It's not inherently problematic that some angles in the hypothetical 80-side polygon are greater than 180 degrees. What is a problem is that one angle in such a polygon would need to be exactly 180 degrees, and you can't have a 180-degree angle in a polygon, because then you have a straight line and do not have a vertex.

There's nothing in the GMAT instructions that tells you to assume all "polygons" are convex (the Math Review section in the OG says something to that effect, but that Math Review is not a substitute for the test directions), but as a practical matter, if you did assume that when a GMAT question mentions a "polygon" that it is a "convex polygon", I think you'll always get the right answer -- I can't imagine seeing a real GMAT question about polygons where the answer would be different if you considered concave polygons separately from convex ones. If you see a polygon with interior angles greater than 180 degrees on the GMAT, the question will include a diagram so you can see what kind of polygon you're dealing with. Q10 in the diagnostic test at the start of the OG is one example (the star shape is concave if you erase the edges of the pentagon).
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The measures of the interior angles in a polygon are consecutive integ [#permalink]
rxs0005 wrote:
The measures of the interior angles in a polygon are consecutive integers. The smallest angle measures 136 degrees. How many sides does this polygon have?

A) 8
B) 9
C) 10
D) 11
E) 13

smallest: a=136; largest: a+(n-1)
sum consec integers: avg (smallest,largest) * n
sum interior angles polygon: 180(n-2)

[a+a+n-1]/2*n=180(n-2)
[136+136+n-1]/2*n=180(n-2)
n^2-89n+720=0
f(720)=720.1,360.2,180.4,90.8,80.9
(n-80)(n-9)=0
n=9

ans (B)
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Re: The measures of the interior angles in a polygon are consecutive integ [#permalink]
mau5 wrote:
enigma123 wrote:
The measures of the interior angles in a polygon are consecutive integers. The smallest angle measures 136 degrees. How many sides does this polygon have?

A) 8
B) 9
C) 10
D) 11
E) 13

Just another way of doing this sum:
The sum of exterior angles for any polygon = 360 degrees.Now, given that the minimum internal angle measure is 136 degrees--> the exterior angle = 180-136 = 44 degrees.
Also, we know that this value will keep decreasing like 43,42,41 etc. It is easy to see that only if there 9 terms, the middle value is 40 and we know that 40*9 =360 degrees.

B.

Can you please explain "It is easy to see that only if there 9 terms, the middle value is 40 and we know that 40*9 =360 degrees."??
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Re: The measures of the interior angles in a polygon are consecutive integ [#permalink]
another way of solving it

since (n-2)*180= the resulting figure has to be zero for any n number of sides.

smallest one is 136 so i kept adding consecutive number till i got zero in unit digit i.e. 136+137+138+139+140+141+142+143+144 = total 9 numbers, hence 9 sides
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Re: The measures of the interior angles in a polygon are consecutive integ [#permalink]
The sum of interior angles will have 0 in its units digits.
Angles are 136, 137, 138, 139, 140...

Sum of consecutive angles must have a 0 at units digits. So, let's only consider the units digits of these numbers.

6+7+8+9+0 = 30 (This means that the answer could be 4 sides of 5 sides as the number of angles = number of sides)
4 or 5 is not in the options

Let's consider a few more angles
141, 142, 143, 144, 145, 146, 147, 148

30+1+2+3+4 = 40 (9 angles - this means B could be correct)
30+1+2+3+4+5 = 45 (11 angles)
30+1+2+3+4+5+6 = 51 (11 angles)
30+1+2+3+4+5+6+7+8 = 66 (13 angles)
So, only B is left
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Re: The measures of the interior angles in a polygon are consecutive integ [#permalink]
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