GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 19 Oct 2019, 05:36

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# If two integers are chosen at random out of the set {2, 5, 7, 8}, what

Author Message
TAGS:

### Hide Tags

Intern
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]

### Show Tags

29 Sep 2018, 06:20
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
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]

### Show Tags

01 Oct 2018, 04:35
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
_________________
Karishma
Veritas Prep GMAT Instructor

Manager
Joined: 08 Jan 2019
Posts: 56
Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what  [#permalink]

### Show Tags

09 May 2019, 18:48
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
Joined: 07 Feb 2018
Posts: 4
Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what  [#permalink]

### Show Tags

10 May 2019, 13:52
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
Joined: 07 Feb 2018
Posts: 4
Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what  [#permalink]

### Show Tags

10 May 2019, 13:52
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
Joined: 07 Feb 2018
Posts: 4
If two integers are chosen at random out of the set {2, 5, 7, 8}, what  [#permalink]

### Show Tags

10 May 2019, 13:55
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

Go to page   Previous    1   2   3   [ 46 posts ]

Display posts from previous: Sort by