Manhattan Prep
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.
OA A
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If Polygon X has fewer than 9 sides, how many sides does
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Brent@GMATPrepNow
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Target question: How many sides does Polygon X have?AAPL wrote:Manhattan Prep
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.
OA A
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
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Brent
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fskilnik@GMATH
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$$\sum\nolimits_N { = \left( {N - 2} \right) \cdot 180\,\,\,\,\,\,\left[ {{\rm{degrees}}} \right]} $$AAPL wrote:Manhattan Prep
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.
$$? = N\,\,\,\,\left( {3 \le \,\,N\,\,{\mathop{\rm int}} \,\, \le 8} \right)$$
$$\left( 1 \right)\,\,\,{\mathop{\rm int}} \,\, = \,\,{{\left( {N - 2} \right) \cdot 180} \over {16}}\,\, = \,\,{{\left( {N - 2} \right) \cdot 45} \over 4}\,\,\,\,\,\mathop \Rightarrow \limits^{GCF\left( {45,4} \right)\, = \,1} \,\,\,\,\,{{N - 2} \over 4} = {\mathop{\rm int}} $$
$$\left. \matrix{
3 \le \,\,N\,\,{\mathop{\rm int}} \,\, \le 8\,\, \hfill \cr
{{N - 2} \over 4} = {\mathop{\rm int}} \hfill \cr} \right\}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,N = 6\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.$$
$$\left( 2 \right)\,\,\,{\mathop{\rm int}} \,\, = \,\,{{\left( {N - 2} \right) \cdot 180} \over {15}}\,\, = \,\,\left( {N - 2} \right) \cdot 12\,\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,N = 3 \hfill \cr
\,{\rm{Take}}\,\,N = 4 \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{INSUFF}}.$$
We follow the notations and rationale taught in the GMATH method.
Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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1) SUFFICIENT: Using the relationship 180(n - 2) = (sum of interior angles), we could calculate the sum of the interior angles for all the polygons that have fewer than 9 sides. Just the first two are shown below; it would take too long to calculate all of the possibilities.
2) INSUFFICIENT: Statement (2) tells us that the sum of the interior angles of Polygon X is divisible by 15. Therefore, the prime factorization of the sum of the interior angles will include 3 × 5. Following the same procedure as above, we realize that both 3 and 5 are included in the prime factorization of 180. As a result, every one of the possibilities can be divided by 15 regardless of the number of sides.
The correct answer is A.
2) INSUFFICIENT: Statement (2) tells us that the sum of the interior angles of Polygon X is divisible by 15. Therefore, the prime factorization of the sum of the interior angles will include 3 × 5. Following the same procedure as above, we realize that both 3 and 5 are included in the prime factorization of 180. As a result, every one of the possibilities can be divided by 15 regardless of the number of sides.
The correct answer is A.
