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# If all members of Set X are positive integers, what is the range of Se

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Math Expert
Joined: 02 Sep 2009
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If all members of Set X are positive integers, what is the range of Se  [#permalink]

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27 Oct 2016, 22:10
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Difficulty:

45% (medium)

Question Stats:

60% (01:48) correct 40% (01:45) wrong based on 100 sessions

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If all members of Set X are positive integers, what is the range of Set X?

(1) The positive difference between the square of the largest number in Set X and the square of the smallest number in Set X is 99.
(2) The sum of the largest number in Set X and the smallest number in Set X is 11.

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Re: If all members of Set X are positive integers, what is the range of Se  [#permalink]

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28 Oct 2016, 02:57
1
Bunuel wrote:
If all members of Set X are positive integers, what is the range of Set X?

(1) The positive difference between the square of the largest number in Set X and the square of the smallest number in Set X is 99.
(2) The sum of the largest number in Set X and the smallest number in Set X is 11.

Let largest number be 'a' and smallest number be 'b'.

a-b?

Statement 1:

a^2 - b^2 = 99

(a+b)(a-b)=99

If a=50 and b =49, then a-b=1;

If a=10, and b =1, then a-b = 9;

Insufficient

Statement 2:

a+b=11,

Clearly insufficient

Statement 1&2:

11*(a-b)=99

So a-b=9

C

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Joined: 11 Sep 2015
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Re: If all members of Set X are positive integers, what is the range of Se  [#permalink]

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14 Jan 2019, 10:58
Top Contributor
Bunuel wrote:
If all members of Set X are positive integers, what is the range of Set X?

(1) The positive difference between the square of the largest number in Set X and the square of the smallest number in Set X is 99.
(2) The sum of the largest number in Set X and the smallest number in Set X is 11.

Target question: What is the range of Set X?
This is a good candidate for rephrasing the target question.
Let G = the Greatest number in set X
Let L = the Least number in set X
REPHRASED target question: What is the value of G - L?

Statement 1: The positive difference between the square of the largest number in Set X and the square of the smallest number in Set X is 99.
In other words, G² - L² = 99
Factor left side: (G + L)(G - L) = 99
IMPORTANT: Using only positive integers, there are only 3 different ways to get a product of 99:
(99)(1) = 99
(33)(3) = 99
(11)(9) = 99

Since G and L are positive INTEGERS, we know that (G + L) is an integer, and (G - L) is an integer
So, only 3 scenarios are possible:
Scenario #1: (G + L) = 99, and (G - L) = 1
Scenario #2: (G + L) = 33, and (G - L) = 3
Scenario #3: (G + L) = 11, and (G - L) = 9

For Scenario #1, the answer to the REPHRASED target question is G - L = 1
For Scenario #2, the answer to the REPHRASED target question is G - L = 3
For Scenario #3, the answer to the REPHRASED target question is G - L = 9
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The sum of the largest number in Set X and the smallest number in Set X is 11
There are several values of G and L that satisfy statement 2. Here are two:
Case a: G = 8 and L = 3. In this case, the answer to the REPHRASED target question is G - L = 5
Case b: G = 10 and L = 1. In this case, the answer to the REPHRASED target question is G - L = 9
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 yielded exactly 3 scenarios:
Scenario #1: (G + L) = 99, and (G - L) = 1
Scenario #2: (G + L) = 33, and (G - L) = 3
Scenario #3: (G + L) = 11, and (G - L) = 9

Statement 2 tells us that G + L = 11
When we examine the possible scenarios, only one (scenario #3) satisfies the condition that G + L = 11
Scenario #3 also tell us that (G - L) = 9
Since we can answer the REPHRASED target question with certainty, the combined statements are SUFFICIENT

Cheers,
Brent
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Re: If all members of Set X are positive integers, what is the range of Se   [#permalink] 14 Jan 2019, 10:58
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