Interior angles problem

[didieravoaka]

Looking for an easy solution for that one too.I know we have to use 180(n-2). How to quickly and easily get to the answer?

Thanks.

Marc

[GMATGuruNY]

If Polygon X has fewer than 9 sides, how many sides does Polygon X have?

(1) The sum of the interior angles of Polygon X is divisible by 16.

(2) The sum of the interior angles of Polygon X is divisible by 15.

The sum of the interior angles of a polygon with n sides = (n-2)(180).

Statement 1: The sum of the interior angles of Polygon X is divisible by 16.
Thus:
(n-2)(180)/16 = positive integer.
(n-2)(45)/4 = positive integer.
Since 4 does not divide into 45, the expression on the left-hand side will yield a positive integer only if (n-2) is a multiple of 4.
If n-2 = 4, then n=6.
If n-2 = 8, then n=10.
Since the question stem requires that n<9, only the option in red is viable.
Thus, n=6.
SUFFICIENT.

Statement 2: The sum of the interior angles of Polygon X is divisible by 15.
Thus:
(n-2)(180)/15 = positive integer.
(n-2)(12) = positive integer.
The product on the left-hand side will yield a positive integer if n is equal to any of the following:
3, 4, 5, 6, 7, 8.
Since n can be different values, INSUFFICIENT.

The correct answer is A.

[didieravoaka]

Thanks Mitch!

[Brent@GMATPrepNow]

If Polygon X has fewer than 9 sides, how many sides does Polygon X have?

(1) The sum of the interior angles of Polygon X is divisible by 16.

(2) The sum of the interior angles of Polygon X is divisible by 15.

Target question: How many sides does Polygon X have?

Given: Polygon X has fewer than 9 sides

Useful rule: The sum of the angles in an n-sided polygon = (n - 2)(180º)

Since the polygon has FEWER than 9 sides, there are are exactly SIX possible cases:
case a: There are 8 sides, in which case the sum of the angles = (8 - 2)(180º) = 6(180º)
case b: There are 7 sides, in which case the sum of the angles = (7 - 2)(180º) = 5(180º)
case c: There are 6 sides, in which case the sum of the angles = (6 - 2)(180º) = 4(180º)
case d: There are 5 sides, in which case the sum of the angles = (5 - 2)(180º) = 3(180º)
case e: There are 4 sides, in which case the sum of the angles = (4 - 2)(180º) = 2(180º)
case f: There are 3 sides, in which case the sum of the angles = (3 - 2)(180º) = 180º

Statement 1: The sum of the interior angles of Polygon X is divisible by 16.
Only case c (6 sides) satisfies this condition.
4(180º) = 720, and 720 is divisible by 16.
Since no other cases satisfy the condition in statement 1, it MUST be the case that the polygon has 6 sides
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The sum of the interior angles of Polygon X is divisible by 15.
Since 180 is divisible by 15, we can be certain that any multiple of 180 is also divisible by 15.
So, cases a through to f all satisfy the condition in statement 2.
In other words, the polygon have have 8, 7, 6, 5, 4, or 3 sides
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer = A

Cheers,
Brent

[Matt@VeritasPrep]

A nice way of remembering this formula is thinking of your polygon in terms of triangles.

A triangle is one triangle, obviously, so it has 180°.

A square is two triangles, so it has 180+180 = 360°.

A pentagon is three triangles, so it has 180+180+180 = 540°, etc.

You can see that the first three sides count ONCE -- they form a ∆ -- and each additional side also counts once. So if we have n sides, the first three count ONCE, the other (n - 3) count once EACH, and we're left with 1 + (n - 3) = n - 2 to count.