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Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  If two integers are chosen at random out of the set {2, 5, 7, 8}, what

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Intern  B
Joined: 16 Feb 2018
Posts: 14
Location: India
Schools: DeGroote'21 (S)
GPA: 3.5
Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what  [#permalink]

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IanStewart wrote:
You can avoid an exhaustive test here. Suppose I ask whether (97)(103) can be written in the form a^2 - b^2, where a and b are integers. Notice that this is a difference of squares: a^2 - b^2 = (a+b)(a-b). We can now just use the median of 97 and 103, which is 100:

(97)(103) = (100-3)(100+3) = 100^2 - 3^2

So whenever we can write our product in such a way that the median of our two numbers is an integer, we can write our product as a difference of squares just as above. For example, if we take 5*7, that's equal to (6-1)(6+1), and if we take 2*8, that's equal to (5-3)(5+3). Now if we look at 8*5, we can't immediately use the same trick, but we can 'move' one of the 2s from the 8 into the 5, as follows: 8*5 = 4*10 = (7-3)(7+3). Similarly, 8*7 = 4*14 = (9-5)(9+5). So of our six possible products, four can be written as a difference of squares.

I am sorry, i still dont know how you got the answer from this. What to do after these steps.
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9706
Location: Pune, India
Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what  [#permalink]

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lostaish wrote:
IanStewart wrote:
You can avoid an exhaustive test here. Suppose I ask whether (97)(103) can be written in the form a^2 - b^2, where a and b are integers. Notice that this is a difference of squares: a^2 - b^2 = (a+b)(a-b). We can now just use the median of 97 and 103, which is 100:

(97)(103) = (100-3)(100+3) = 100^2 - 3^2

So whenever we can write our product in such a way that the median of our two numbers is an integer, we can write our product as a difference of squares just as above. For example, if we take 5*7, that's equal to (6-1)(6+1), and if we take 2*8, that's equal to (5-3)(5+3). Now if we look at 8*5, we can't immediately use the same trick, but we can 'move' one of the 2s from the 8 into the 5, as follows: 8*5 = 4*10 = (7-3)(7+3). Similarly, 8*7 = 4*14 = (9-5)(9+5). So of our six possible products, four can be written as a difference of squares.

I am sorry, i still dont know how you got the answer from this. What to do after these steps.

Check out this post: https://www.veritasprep.com/blog/2014/0 ... at-part-i/

A number can be written in the form a^2 - b^2 if it is odd or has 4 as a factor (explained in the post above)

There are 4C2 = 6 ways of picking a pair of numbers here. Out of these, in only two cases (2, 5) and (2, 7) you will not be able to write the product as a^2 - b^2. Since product will not be odd and will not have 4 as a factor.
So required probability = 4/6 = 2/3
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Manager  B
Joined: 08 Jan 2019
Posts: 56
Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what  [#permalink]

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Convoluted approach with hugely complicated calculations - looks like a easy breezy vacation solution rather than a GMAT solution. Combination approach would be much faster in this case.
Intern  B
Joined: 07 Feb 2018
Posts: 4
Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what  [#permalink]

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MitDavidDv wrote:
If two integers are chosen at random out of the set {2, 5, 7, 8}, what is the probability that their product will be of the form a^2 – b^2, where a and b are both positive integers?

A. 2/3
B. 1/2
C. 1/3
D. 1/4
E. 1/6

Shalom! I am currently studying the probability chapter of the Manhattan GMAT Word Translations book. I am looking forward to the different outcomes and answers.

I did the same thing but failed to include 8*5 and 8*7, thanks by the way.
Intern  B
Joined: 07 Feb 2018
Posts: 4
Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what  [#permalink]

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MitDavidDv wrote:
If two integers are chosen at random out of the set {2, 5, 7, 8}, what is the probability that their product will be of the form a^2 – b^2, where a and b are both positive integers?

A. 2/3
B. 1/2
C. 1/3
D. 1/4
E. 1/6

Shalom! I am currently studying the probability chapter of the Manhattan GMAT Word Translations book. I am looking forward to the different outcomes and answers.

I did the same thing but failed to include 8*5 and 8*7, thanks by the way.
Intern  B
Joined: 07 Feb 2018
Posts: 4
If two integers are chosen at random out of the set {2, 5, 7, 8}, what  [#permalink]

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lostaish wrote:
IanStewart wrote:
You can avoid an exhaustive test here. Suppose I ask whether (97)(103) can be written in the form a^2 - b^2, where a and b are integers. Notice that this is a difference of squares: a^2 - b^2 = (a+b)(a-b). We can now just use the median of 97 and 103, which is 100:

(97)(103) = (100-3)(100+3) = 100^2 - 3^2

So whenever we can write our product in such a way that the median of our two numbers is an integer, we can write our product as a difference of squares just as above. For example, if we take 5*7, that's equal to (6-1)(6+1), and if we take 2*8, that's equal to (5-3)(5+3). Now if we look at 8*5, we can't immediately use the same trick, but we can 'move' one of the 2s from the 8 into the 5, as follows: 8*5 = 4*10 = (7-3)(7+3). Similarly, 8*7 = 4*14 = (9-5)(9+5). So of our six possible products, four can be written as a difference of squares.

I am sorry, i still dont know how you got the answer from this. What to do after these steps.

After these steps you will get 4 products in the form of a^2-b^2, and we have total products 6 (4C2),
probability will be 4/6=2/3 Ans Which is Option B. If two integers are chosen at random out of the set {2, 5, 7, 8}, what   [#permalink] 10 May 2019, 13:55

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