Author 
Message 
Director
Joined: 01 Jan 2008
Posts: 602

X is called a triangular number if X = 1 + 2 + ... + n for [#permalink]
Show Tags
01 Oct 2008, 06:15
1
This post received KUDOS
X is called a triangular number if X = 1 + 2 + ... + n for some positive integer n X is called a perfect square if X = k^2 for some integer k The first X which is both a triangular number and a perfect square is 1, the second 36. What's the third one? == Message from GMAT Club Team == This is not a quality discussion. It has been retired. If you would like to discuss this question please repost it in the respective forum. Thank you! To review the GMAT Club's Forums Posting Guidelines, please follow these links: Quantitative  Verbal Please note  we may remove posts that do not follow our posting guidelines. Thank you.



Manager
Joined: 22 Sep 2008
Posts: 117

Re: walker type of problem [#permalink]
Show Tags
01 Oct 2008, 06:46
i think we should try this type of question by OPTION only..
Ultimately in gmat we are going to have option , just apply 4 option one by one , we will get answer definately
* i think we need to find number which is cube of some number and following number is square of some number.



Intern
Joined: 29 Sep 2008
Posts: 46

Re: walker type of problem [#permalink]
Show Tags
01 Oct 2008, 07:59
1
This post received KUDOS
a number that satisfies both requirements can be expressed as n(n+1)/2 and k^2.
Since it is the same number:
n(n+1)/2=k^2 => n(n+1) = 2 * k^2
careful observation reveals that we are looking for a positive integer that can be expressed as a product of two consecutive positive integers and also as double of a perfect square.
quick things to realize: 1. two consecutive positive integers would never have a common factor (except 1 which we can ignore) 2. two consecutive positive integers would have one odd and one even integer 3. odd integer has to be a perfect square 4. 2 is a factor of only the even integer which would cancel out the 2 from the other side of the equation 5. after the factor 2 is taken out of the even integer, it is also a perfect square
now lets start looking for odd perfect squares whose adjacent even integer is double of a perfect square.
1 and 2  our number is 1 * 2 /2 = 1 9 and 8  our number is 9 * 8 /2 = 36 25  reject, 24 or 26 do not satisfy 49 and 50  our number is 49 * 50 /2 = 1225



Director
Joined: 01 Jan 2008
Posts: 602

Re: walker type of problem [#permalink]
Show Tags
01 Oct 2008, 08:14
Nice solution, aim2010. I solved it exactly the same way.



Manager
Joined: 30 Sep 2008
Posts: 111

Re: walker type of problem [#permalink]
Show Tags
01 Oct 2008, 08:22
aim2010 wrote: a number that satisfies both requirements can be expressed as n(n+1)/2 and k^2.
Since it is the same number:
n(n+1)/2=k^2 => n(n+1) = 2 * k^2
careful observation reveals that we are looking for a positive integer that can be expressed as a product of two consecutive positive integers and also as double of a perfect square.
quick things to realize: 1. two consecutive positive integers would never have a common factor (except 1 which we can ignore) 2. two consecutive positive integers would have one odd and one even integer 3. odd integer has to be a perfect square 4. 2 is a factor of only the even integer which would cancel out the 2 from the other side of the equation 5. after the factor 2 is taken out of the even integer, it is also a perfect square
now lets start looking for odd perfect squares whose adjacent even integer is double of a perfect square.
1 and 2  our number is 1 * 2 /2 = 1 9 and 8  our number is 9 * 8 /2 = 36 25  reject, 24 or 26 do not satisfy 49 and 50  our number is 49 * 50 /2 = 1225 for these similar questions, I won't go so far, esp to many calculations, up to 49(!!!), should we just replace the number from the choices and check n(n+1) = 2 * k^2 if n is integer



Intern
Joined: 29 Sep 2008
Posts: 46

Re: walker type of problem [#permalink]
Show Tags
01 Oct 2008, 08:50
lylya4 wrote: for these similar questions, I won't go so far, esp to many calculations, up to 49(!!!), should we just replace the number from the choices and
check n(n+1) = 2 * k^2 if n is integer
it makes sense to a certain extent, though there are two reasons I would not test answer choices: 1. Imagine checking 5 four digit numbers if they are perfect squares and if all of them are, then doubling them and checking their factors to come up with two consecutive integers as two exhaustive factors 2. checking answer choices at times gives me a feeling that maybe i missed something, what if i do a mistake and two choices seem to satisfy. moreover, for going till 49 we just went till 7 after counting 1, 3 and 5. remember we were looking for odd perfect squares. IMO, its up to each person's choice. bottom line is to not bang your head on something for 4 minutes unless you have a gut feeling that you are going in the right direction.



CEO
Joined: 17 Nov 2007
Posts: 3486
Concentration: Entrepreneurship, Other
Schools: Chicago (Booth)  Class of 2011

Re: walker type of problem [#permalink]
Show Tags
01 Oct 2008, 14:10
Thanks maraticus for a good question and aim2010 for a nice explanation +1+1
_________________
HOT! GMAT TOOLKIT 2 (iOS) / GMAT TOOLKIT (Android)  The OFFICIAL GMAT CLUB PREP APP, a musthave app especially if you aim at 700+  PrepGame



Manager
Joined: 30 Sep 2008
Posts: 111

Re: walker type of problem [#permalink]
Show Tags
01 Oct 2008, 16:25
aim2010 wrote: lylya4 wrote: for these similar questions, I won't go so far, esp to many calculations, up to 49(!!!), should we just replace the number from the choices and
check n(n+1) = 2 * k^2 if n is integer
it makes sense to a certain extent, though there are two reasons I would not test answer choices: 1. Imagine checking 5 four digit numbers if they are perfect squares and if all of them are, then doubling them and checking their factors to come up with two consecutive integers as two exhaustive factors 2. checking answer choices at times gives me a feeling that maybe i missed something, what if i do a mistake and two choices seem to satisfy. moreover, for going till 49 we just went till 7 after counting 1, 3 and 5. remember we were looking for odd perfect squares. IMO, its up to each person's choice. bottom line is to not bang your head on something for 4 minutes unless you have a gut feeling that you are going in the right direction. i still think its easier to check the choices remember k^2 = X So n(n+1) = 2X <=> n^2 + 2n  2X = 0, you replace X with the answer choice and solve the equation, if n is integer, take X much faster



VP
Joined: 18 May 2008
Posts: 1182

Re: walker type of problem [#permalink]
Show Tags
01 Oct 2008, 17:38
That seems to be a better approach lylya4 wrote: aim2010 wrote: lylya4 wrote: for these similar questions, I won't go so far, esp to many calculations, up to 49(!!!), should we just replace the number from the choices and
check n(n+1) = 2 * k^2 if n is integer
it makes sense to a certain extent, though there are two reasons I would not test answer choices: 1. Imagine checking 5 four digit numbers if they are perfect squares and if all of them are, then doubling them and checking their factors to come up with two consecutive integers as two exhaustive factors 2. checking answer choices at times gives me a feeling that maybe i missed something, what if i do a mistake and two choices seem to satisfy. moreover, for going till 49 we just went till 7 after counting 1, 3 and 5. remember we were looking for odd perfect squares. IMO, its up to each person's choice. bottom line is to not bang your head on something for 4 minutes unless you have a gut feeling that you are going in the right direction. i still think its easier to check the choices remember k^2 = X So n(n+1) = 2X n^2 + 2n  2X = 0, you replace X with the answer choice and solve the equation, if n is integer, take X much faster == Message from GMAT Club Team == This is not a quality discussion. It has been retired. If you would like to discuss this question please repost it in the respective forum. Thank you! To review the GMAT Club's Forums Posting Guidelines, please follow these links: Quantitative  Verbal Please note  we may remove posts that do not follow our posting guidelines. Thank you.




Re: walker type of problem
[#permalink]
01 Oct 2008, 17:38






