Jun 29 07:00 AM PDT  09:00 AM PDT Learn reading strategies that can help even nonvoracious reader to master GMAT RC Jun 30 07:00 AM PDT  09:00 AM PDT Get personalized insights on how to achieve your Target Quant Score. Jul 01 08:00 AM PDT  09:00 AM PDT Game of Timers is a teambased competition based on solving GMAT questions to win epic prizes! Starting July 1st, compete to win prep materials while studying for GMAT! Registration is Open! Jul 01 10:00 PM PDT  11:00 PM PDT Join a FREE 1day workshop and learn how to ace the GMAT while keeping your fulltime job. Limited for the first 99 registrants.
Author 
Message 
TAGS:

Hide Tags

Manager
Joined: 28 Jan 2011
Posts: 76
Location: Tennessee
Schools: Belmont University

If two integers are chosen at random out of the set {2, 5, 7, 8}, what
[#permalink]
Show Tags
Updated on: 11 Nov 2017, 23:32
Question Stats:
21% (02:46) correct 79% (02:44) wrong based on 2521 sessions
HideShow timer Statistics
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.
Official Answer and Stats are available only to registered users. Register/ Login.
Originally posted by MitDavidDv on 02 Jun 2011, 11:04.
Last edited by Bunuel on 11 Nov 2017, 23:32, edited 2 times in total.
Renamed the topic, edited the question and added the OA.




GMAT Tutor
Joined: 24 Jun 2008
Posts: 1663

Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what
[#permalink]
Show Tags
03 Jun 2011, 18:50
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)(ab). We can now just use the median of 97 and 103, which is 100: (97)(103) = (1003)(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 (61)(6+1), and if we take 2*8, that's equal to (53)(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 = (73)(7+3). Similarly, 8*7 = 4*14 = (95)(9+5). So of our six possible products, four can be written as a difference of squares.
_________________
GMAT Tutor in Toronto
If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com




Retired Moderator
Joined: 20 Dec 2010
Posts: 1743

Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what
[#permalink]
Show Tags
03 Jun 2011, 12:49
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 just used the exhaustive method. Count everything that fits. 2,5=10 2,7=14 2,8=16 5,7=35 5,8=40 7,8=56 Write down all perfect squares until 100 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 Pick one number at a time. 10  Keep adding with every perfect square and see whether the result is also there in the set. 10+1=11(Not there) 10+4=14(Not there) 10+9=19(Not there) 10+16=26(Not there) 10+25=35(Not there) we can stop here as the difference between all consecutive perfect squares after 35 will be more than 10. 10 Not possible to represented as a^2b^2 Repeat the same with all the products; 14Not Possible 16: Let's check this; 16+1=17(Not there) 16+4=20(Not there) 16+9=25(There in the set) 16 can be represented as a^2b^2 i.e. 5^23^2 ********************************** Likewise: 35 6^21^2 40 7^23^2 56 9^25^2 ******************* In the sample set: {10,14,16,35,40,56} {10,14} Not Possible: Count=2 {16,35,40,56} Possible: Count=4 Total Count=6 P=Favorable/Total=4/6=2/3 Ans: "A" **************************
_________________




Intern
Joined: 25 Sep 2010
Posts: 48
Schools: HBS, LBS, Wharton, Kelloggs, Booth

Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what
[#permalink]
Show Tags
20 Jul 2011, 00:06
IanStewart you gave a very easy explanation! Thank you! +1 kudos



Intern
Joined: 07 Mar 2013
Posts: 5

Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what
[#permalink]
Show Tags
02 Oct 2013, 01:26
Some Theory here: Consider two numbers a, b Now a* b always = [(a+b)/2]^ 2 – [(ab)/2]^2 eqn 1 Additionally: The number of ways a number can be expressed as a difference of two integers depends on number of ways it can be written as a two factor product stated below. a.odd*odd b. even *even The reason being in eqn (1) above [(ab)/2]^2 should result in an integer . Hence we consider only the above set of two factor products. (0dd minus odd = even ,even minus even =even , hence both will be divisible by 2
E.g the number 36 can be written as 6*6 , 18*2 hence 36 can be expressed as difference of squares in two ways. 36 = (6+6)/2 ^ 2  (66)/2 ^ 2 = 6^2  0 36= (18+2)/2^2 – (182)/^2 = 10^2 – 4^2
Now back to the problem : We have the number set (2, 5, 7, 8) out of which, the satisfying possibilities as per the above theory would be 2*8 (valid) 5*7 (valid) 2*5 (not valid) 2*7(not valid) 5*8== 4*10 hence valid 7*8 == 4*14 hence valid) Hence there are 4 favourble cases out of 6. Therefore probability Is 4/6 = 2/3



Manager
Joined: 06 Feb 2010
Posts: 145
Concentration: Marketing, Leadership
Schools: University of Dhaka  Class of 2010
GPA: 3.63
WE: Business Development (Consumer Products)

Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what
[#permalink]
Show Tags
28 Oct 2013, 21:20
Need Bunuel's explanation for this problem......
_________________
Practice Makes a Man Perfect. Practice. Practice. Practice......Perfectly
Critical Reasoning: http://gmatclub.com/forum/bestcriticalreasoningshortcutsnotestips91280.html
Collections of MGMAT CAT: http://gmatclub.com/forum/collectionsofmgmatcatmath152750.html
MGMAT SC SUMMARY: http://gmatclub.com/forum/mgmatscsummaryoffourthedition152753.html
Sentence Correction: http://gmatclub.com/forum/sentencecorrectionstrategiesandnotes91218.html
Arithmatic & Algebra: http://gmatclub.com/forum/arithmaticalgebra93678.html
Helpful Geometry formula sheet: http://gmatclub.com/forum/bestgeometry93676.html
I hope these will help to understand the basic concepts & strategies. Please Click ON KUDOS Button.



Math Expert
Joined: 02 Sep 2009
Posts: 55804

Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what
[#permalink]
Show Tags
29 Oct 2013, 02:29
monirjewel wrote: Need Bunuel's explanation for this problem...... Best solution is here: iftwointegersarechosenatrandomoutoftheset114579.html#p929326
_________________



Senior Manager
Joined: 03 Apr 2013
Posts: 273
Location: India
Concentration: Marketing, Finance
GPA: 3

Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what
[#permalink]
Show Tags
17 Nov 2013, 10:35
Bunuel wrote: monirjewel wrote: Need Bunuel's explanation for this problem...... Best solution is here: iftwointegersarechosenatrandomoutoftheset114579.html#p929326 Bunuel please help out here..this will clear many things for me. My solution is this : a^2  b^2 will be of the form (ab)(a+b). We can infer that the difference between the two chosen numbers(of which one is (ab) and the other (a+b)) will be (a+b)  (ab) = 2b i.e. even difference(negative or positive). Thus we will have to choose either two even numbers or two odd numbers? this way we have two options > 1. choosing 5 and 7 of which probability is = 1/6 2. choosing 2 and 8 of which probability is = 1/6. thus the total probability = 2/6 which is 1/3. please explain why this is wrong.
_________________
Spread some love..Like = +1 Kudos



Intern
Joined: 27 Feb 2014
Posts: 17

Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what
[#permalink]
Show Tags
27 Feb 2014, 23:13
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 have an explanation too, maybe it'd be of some help: a^2  b^2 =(ab)(a+b) (a+b) and (ab) can only be integers from the selected set i.e {2,5,7,8} Now a and b are both positive integers as stated in the question So the sum of (ab) and (a+b) also has to be an integer i.e 2a = sum of any two numbers chosen from the set {2,5,7,8} For a to be a positive integer the sum has to be an even number so either it could be a pair of (2,8) or (7,5) Using PnC P(getting one such pair when chosing two random numbers from a set) = P(Both numbers chosen to be even) + P(Both numbers chosen to be odd) = 2/4C2 + 2/4C2 = 4/4C2 = 4/6 = 2/3 Kindly let me know in case the solution has some mistakes.



Retired Moderator
Joined: 20 Dec 2013
Posts: 171
Location: United States (NY)
GMAT 1: 640 Q44 V34 GMAT 2: 710 Q48 V40 GMAT 3: 720 Q49 V40
GPA: 3.16
WE: Consulting (Venture Capital)

Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what
[#permalink]
Show Tags
28 Feb 2014, 22:04
a^2b^2 = (a+b)(ab), so if the difference of any the factor pairs of the product is a positive even integer, then they can be in the a^2b^2 form. Example: 7*5=35 > 7*5 = (6+1)(61) > 35*1 = (1817)(18+17), etc etc Only products that don't work are 2*5=10 (as 53 = odd and 101 = odd) and 2*7=14 (as 75 = odd and 141=13 odd). The other 4 products work and so the probability = 4/6 = 2/3
_________________



Intern
Joined: 21 Aug 2013
Posts: 6

Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what
[#permalink]
Show Tags
08 Apr 2014, 04:43
ShashankDave wrote: Bunuel wrote: monirjewel wrote: Need Bunuel's explanation for this problem...... Best solution is here: iftwointegersarechosenatrandomoutoftheset114579.html#p929326 Bunuel please help out here..this will clear many things for me. My solution is this : a^2  b^2 will be of the form (ab)(a+b). We can infer that the difference between the two chosen numbers(of which one is (ab) and the other (a+b)) will be (a+b)  (ab) = 2b i.e. even difference(negative or positive). Thus we will have to choose either two even numbers or two odd numbers? this way we have two options > 1. choosing 5 and 7 of which probability is = 1/6 2. choosing 2 and 8 of which probability is = 1/6. thus the total probability = 2/6 which is 1/3. please explain why this is wrong. I had the same concern but now it's clear for me : the 6 possible pairs are (2 5) (2 7) (2 8) (5 7) (5 8) (7 8) our method allows to find 2 pairs (2 8) (5 7) but does not allow to eliminate the others Especially (5 8 ) and (7 8) that also meet the condition : 5*8 = 10*4 = (7+3)*(73) and 7*8=14*4=(9+5)*(95) so there are 4 possible pairs to be picked from 6 hence the probability is 4/6 = 2/3



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9369
Location: Pune, India

Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what
[#permalink]
Show Tags
13 Apr 2014, 20:09
ShashankDave wrote: Bunuel wrote: monirjewel wrote: Need Bunuel's explanation for this problem...... Best solution is here: iftwointegersarechosenatrandomoutoftheset114579.html#p929326 Bunuel please help out here..this will clear many things for me. My solution is this : a^2  b^2 will be of the form (ab)(a+b). We can infer that the difference between the two chosen numbers(of which one is (ab) and the other (a+b)) will be (a+b)  (ab) = 2b i.e. even difference(negative or positive). Thus we will have to choose either two even numbers or two odd numbers? this way we have two options > 1. choosing 5 and 7 of which probability is = 1/6 2. choosing 2 and 8 of which probability is = 1/6. thus the total probability = 2/6 which is 1/3. please explain why this is wrong. The question is not so much as whether both the numbers are even or both are odd as whether the product of the numbers can be written as product of two even numbers or two odd numbers. Two numbers are chosen and multiplied. Now they have lost their individual identity. Now you focus on the product and find whether it can be written as product of two numbers which are both odd or both even. Say you took two number 7 and 8 and multiplied them. You get 56. Can you write 56 as product of two numbers such that both are even? Yes, 4 and 14 or 2 and 28. So 56 can be written as a^2  b^2 in two ways: (9^2  5^2) and (15^2  13^3). So if you choose 7, 8 from the set, their product can be written in the form a^2  b^2. Similarly, 5, 8 will give you the same result. Hence you get 2 more cases and total probability becomes 4/6 = 2/3. Whenever you have at least 4 in the product, you can write it as product of two even numbers: give one 2 to one number and the other 2 to the other number to make both even. If the product is even but not a multiple of 4, it cannot be written as product of two even numbers or product of two odd numbers. It can only be written as product of one even and one odd number. If the product is odd, it can always be written as product of two odd numbers.
_________________
Karishma Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >



Intern
Joined: 05 Aug 2014
Posts: 1

Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what
[#permalink]
Show Tags
05 Aug 2014, 13:31
Here is the solution with easy steps:
a^2  b^2 = (ab)(a+b)
Now we need to find all possible combinations of two numbers from the set {2, 5, 7, 8 } which can be expressed as (ab)(a+b)
Let say x =ab and y = a+b, therefore x+y = 2a and yx = 2b, so you need to have two numbers x and y whose sum and difference should be even number.
How many are there from the set {2, 5, 7, 8} ? 8 + 2 = 10 , 8  2 = 6, 7  5 = 2 , 7 + 5 = 12 . So there are two pairs (8,2) and (7,5) which can be expressed as a^2  b^2.
Therefore, the probability that their product will be of the form a^2 – b^2 = 2/total two numbers combination = 2/4 Chose 2 = 2/3
Ans is option (A)



Senior Manager
Joined: 08 Apr 2012
Posts: 344

Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what
[#permalink]
Show Tags
13 Sep 2014, 11:12
VeritasPrepKarishma wrote: The question is not so much as whether both the numbers are even or both are odd as whether the product of the numbers can be written as product of two even numbers or two odd numbers.
Two numbers are chosen and multiplied. Now they have lost their individual identity. Now you focus on the product and find whether it can be written as product of two numbers which are both odd or both even. Say you took two number 7 and 8 and multiplied them. You get 56. Can you write 56 as product of two numbers such that both are even? Yes, 4 and 14 or 2 and 28. So 56 can be written as a^2  b^2 in two ways: (9^2  5^2) and (15^2  13^3). So if you choose 7, 8 from the set, their product can be written in the form a^2  b^2.
Similarly, 5, 8 will give you the same result.
Hence you get 2 more cases and total probability becomes 4/6 = 2/3.
Whenever you have at least 4 in the product, you can write it as product of two even numbers: give one 2 to one number and the other 2 to the other number to make both even. If the product is even but not a multiple of 4, it cannot be written as product of two even numbers or product of two odd numbers. It can only be written as product of one even and one odd number. If the product is odd, it can always be written as product of two odd numbers.
Hi Karishma, Can you elaborate a little more? Why are we looking for 2 numbers that are either both even or both odd? Also, how did you know to stop at (9^2  5^2) and (15^2  13^3) and not look for more? Thanks,



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9369
Location: Pune, India

Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what
[#permalink]
Show Tags
15 Sep 2014, 02:21
ronr34 wrote: Hi Karishma, Can you elaborate a little more? Why are we looking for 2 numbers that are either both even or both odd? Also, how did you know to stop at (9^2  5^2) and (15^2  13^3) and not look for more? Thanks, That's a good question. You should understand this concept well. That is why I have written a detailed post on it on my blog: http://www.veritasprep.com/blog/2014/04 ... atparti/Check it out and get back to me (on the blog or here) if any doubts remain.
_________________
Karishma Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >



Intern
Joined: 07 Sep 2014
Posts: 17
Location: United States (MA)
Concentration: Finance, Economics

Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what
[#permalink]
Show Tags
20 Oct 2014, 07:56
not sure if its been discussed, but a valuable property to know is that ANY nonprime odd number, or multiple of 4, can be written as a difference of squares using integers. 21 = (5+2)(52) 15 = (4+1)(41) etc. try it out.
therefore, we can see that out of our 6 possible outcomes, only 4 will be either odd (5 x 7) or multiples of 4 (8 x each other #). so answer = 4/6=2/3



Manager
Joined: 11 Nov 2011
Posts: 62
Location: United States
Concentration: Finance, Human Resources
GPA: 3.33
WE: Consulting (NonProfit and Government)

Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what
[#permalink]
Show Tags
14 Jan 2015, 19:05
Very simple and straightforward method but never heard about the number property.... bsmith37 wrote: not sure if its been discussed, but a valuable property to know is that ANY nonprime odd number, or multiple of 4, can be written as a difference of squares using integers. 21 = (5+2)(52) 15 = (4+1)(41) etc. try it out.
therefore, we can see that out of our 6 possible outcomes, only 4 will be either odd (5 x 7) or multiples of 4 (8 x each other #). so answer = 4/6=2/3



Director
Joined: 07 Aug 2011
Posts: 518
Concentration: International Business, Technology

Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what
[#permalink]
Show Tags
13 Mar 2015, 21:17
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. So far if the average of the two numbers is an INTEGER they can be written in (a+b)(ab) form . so that narrows us down to Odd + Odd and Even+Even cases . Special consideration need to taken for those cases in which one number is ODD and other is multiple of 4 , i.e. in this case if the set is \({ 2,5,7,8 }\), then possible pairs are : 7*8 = 56 = 14*4 = 28*2 none of these pairs (7,8) , (14,4), and (28*2) can be expressed in (a+b) (ab) form . 5*8= 40 = 10*4 = (7+3) (73), so yes we can write \(5*8\) as \((7+3) * (73)\) 2,5,7,8 total number of cases = 4C2 = 6 favorable cases : (odd,odd) (5,7) , (Even,Even) (2,8) , and one special case as shown above (5,8) so \(3/6=1/2\)



Manager
Joined: 10 Jun 2015
Posts: 117

Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what
[#permalink]
Show Tags
11 Jun 2015, 22:19
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. answer is (A) the product set=(10, 14, 16, 35, 40, and 56) 16=8x2=(5+3)(53); (82)/2 =3 35=7x5=(6+1)(61); (75)/2 = 1 40=10x4=(7+3)(73) 56=14x4=(9+5)(95) you got the pattern.



Intern
Joined: 05 Mar 2014
Posts: 6

Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what
[#permalink]
Show Tags
05 Sep 2015, 20:45
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 consider AxB = a^2 – b^2 = (a+b)(ab) where A and B are the posible chosen first: the total posible chosen is 12, because A take 4 values and B takes 3, 4x3 = 12 Second This is the scenary: A x B = (a+b)(ab) 2 5 = 10 = 2x5 or 10x1 (wrong) note that the sum of the factors should be even number, (conditions from a and b are integers), for the reason these opstion is eliminated 2 7 = 14 = 7x2 (wrong, the sum is not even) or 14x1 (wrong) 2 8 = 16 = 8x2 (correct) 5 2 = 10 = 8x2 (wrong, and the same that the first) 5 7 = 35 = 5x7 (Correct) 5 8 = 40 = 20x2 (correct) 7 2 = 14 = (wrong and is the same ) 7 5 = 35 = 7x5 (correct) 7 8 = 56 = 14x4 (correct) 8 2 = 16 = 8x2 (correct) 8 5 = 40 = 20x2 (correct) 8 7 = 56 = 14x4 (correct) Finally, the number of correct posible answer is 8, and the total possible answer is 12 indeed, 8/12 = 2/3




Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what
[#permalink]
05 Sep 2015, 20:45



Go to page
1 2 3
Next
[ 46 posts ]



