Author 
Message 
TAGS:

Hide Tags

Manager
Joined: 25 Aug 2011
Posts: 133
Location: India
WE: Operations (Insurance)

If a is the sum of x consecutive positive integers. b is the
[#permalink]
Show Tags
Updated on: 06 Mar 2012, 02:50
Question Stats:
30% (02:41) correct 70% (02:26) wrong based on 589 sessions
HideShow timer Statistics
If a is the sum of x consecutive positive integers. b is the sum of y consecutive positive integers. For which of the following values of x and y is it impossible that a = b? A. x = 2; y = 6 B. x = 3; y = 6 C. x = 7; y = 9 D. x = 10; y = 4 E. x = 10; y = 7
Official Answer and Stats are available only to registered users. Register/ Login.
Originally posted by devinawilliam83 on 05 Mar 2012, 23:36.
Last edited by Bunuel on 06 Mar 2012, 02:50, edited 2 times in total.
Edited the question




Math Expert
Joined: 02 Sep 2009
Posts: 65761

Re: sum of x consecutive positive integers
[#permalink]
Show Tags
06 Mar 2012, 02:48
If a is the sum of x consecutive positive integers. b is the sum of y consecutive positive integers. For which of the following values of x and y is it impossible that a = b?A. x = 2; y = 6 B. x = 3; y = 6 C. x = 7; y = 9 D. x = 10; y = 4 E. x = 10; y = 7 The sum n consecutive integers is give by: \(Sum=\frac{(2a_1+n1)*n}{2}\) (check Number Theory chapter of Math Book for more: mathnumbertheory88376.html); Notice that: If \(n=even=2*odd\), so when \(n\) (# of consecutive integers) is even but not a multiple 4 then \(Sum=\frac{(2a_1+n1)*n}{2}=\frac{(even+evenodd)*(2*odd)}{2}=odd*odd=odd\); If \(n=even=2*even\), so when \(n\) is a multiple 4 then \(Sum=\frac{(2a_1+n1)*n}{2}=\frac{(even+evenodd)*(2*even)}{2}=odd*even=even\); That's because a set of even number of consecutive integers has half even and half odd terms. The sum of even terms is obviously even. As for odd terms: their sum is even if their number is even (so total # of terms is multiple of 4) and their sum is odd if their number is odd (so total number of terms is even but not a multiple of 4); So, the sum of 10 (not a multiple of 4) consecutive integers will be odd (the sum of 5 even and 5 odd integers) and the sum of 4 (multiple of 4) consecutive integers will be even (the sum of 2 even and 2 odd integers), so option D is not possible. Answer: D. Hope it's clear.
_________________




Manager
Joined: 20 Dec 2013
Posts: 114

Re: If a is the sum of x consecutive positive integers. b is the
[#permalink]
Show Tags
11 Apr 2014, 03:53
devinawilliam83 wrote: If a is the sum of x consecutive positive integers. b is the sum of y consecutive positive integers. For which of the following values of x and y is it impossible that a = b?
A. x = 2; y = 6 B. x = 3; y = 6 C. x = 7; y = 9 D. x = 10; y = 4 E. x = 10; y = 7 Sum of 2 consecutive integers = Odd Sum of 4 consecutive integers = Even Sum of 6 consecutive integers = Odd Sum of 8 consecutive integer = Even and so on is the pattern. If we look at answer option D. Sum of 10 consecutive integer = Odd, Sum of 4 consecutive integers = Even. Odd cannot be equal to Even. Hence the answer is D.
_________________
76000 Subscribers, 7 million minutes of learning delivered and 5.6 million video views
Perfect Scores http://perfectscores.org http://www.youtube.com/perfectscores




Manager
Joined: 25 Aug 2011
Posts: 133
Location: India
WE: Operations (Insurance)

Re: If a is the sum of x consecutive positive integers. b is the
[#permalink]
Show Tags
06 Mar 2012, 22:48
Hi, I am struggling with the explanation.Here is what I had done. but was unable to eliminate answers the premise was that for x and y to be equal both should be either even or odd Please take a look and let me know
Attachments
odd even.png [ 6.53 KiB  Viewed 26780 times ]



Math Expert
Joined: 02 Sep 2009
Posts: 65761

Re: If a is the sum of x consecutive positive integers. b is the
[#permalink]
Show Tags
07 Mar 2012, 00:46
devinawilliam83 wrote: Hi, I am struggling with the explanation.Here is what I had done. but was unable to eliminate answers the premise was that for x and y to be equal both should be either even or odd Please take a look and let me know I'm not sure what you are trying to say there with the diagram but the 4th row is not correct: if the # of terms is multiple of 4 then the sum is even, regardless of the first term.
_________________



SVP
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2447
Location: Malaysia
Concentration: Technology, Entrepreneurship
GMAT 1: 670 Q49 V31 GMAT 2: 710 Q50 V35

Re: If a is the sum of x consecutive positive integers. b is the
[#permalink]
Show Tags
27 Oct 2012, 20:25
For consecutive terms , Sum = median * number of terms. So if m1 and m2 are two medians.. a= m1*x and b = 2*y as per given condition. a=b => m1*x = m2*y if you put x=10 and y =4 10m1 = 4m2 => 5m1 = 2m2..... When number of terms are even, the median is always a fraction. I.e 3.5 or 4.5 but when number of terms is odd then median is always an integer. Now in above case. m1 and m2 both are fractions but m2's fraction can be nullified by 2 which is multiplied by m2. But for m1 it is not the case. So all combinations are possible but if number of terms are multiple of 4, then the other can not be of the form 4n+1 or 4n+2 or 4n+3. Hence D
_________________
Fight for your dreams : For all those who fear from Verbal lets give it a fightMoney Saved is the Money Earned Jo Bole So Nihaal , Sat Shri Akaal Support GMAT Club by putting a GMAT Club badge on your blog/Facebook GMAT Club Premium Membership  big benefits and savingsGmat test review : http://gmatclub.com/forum/670to710alongjourneywithoutdestinationstillhappy141642.html



Intern
Joined: 06 Apr 2011
Posts: 13

Re: If a is the sum of x consecutive positive integers. b is the
[#permalink]
Show Tags
02 Feb 2013, 10:05
Toooo good Bunuel... Instead of learning formula i am more comfortable with the conceptual approach. Do you think learning formula is integral for GMAT.



Verbal Forum Moderator
Joined: 10 Oct 2012
Posts: 562

Re: If a is the sum of x consecutive positive integers. b is the
[#permalink]
Show Tags
03 Feb 2013, 02:21
We know that if x=10, and say the first term is odd, then the sum(a) will be 5x(odd+even) = 5x(odd)= odd. The first term can be even also, nonetheless, the sum will be 5x(even+odd)=5x(odd)= Still odd. Now for y=4, whatever the first term be,(odd/even) the sum b = 2(odd+even) = 2x(odd)=even. Thus as odd cant be equal to even, ans is D.
_________________



Intern
Joined: 06 Aug 2012
Posts: 15

Re: sum of x consecutive positive integers
[#permalink]
Show Tags
30 Nov 2013, 06:43
Bunuel wrote: If a is the sum of x consecutive positive integers. b is the sum of y consecutive positive integers. For which of the following values of x and y is it impossible that a = b?A. x = 2; y = 6 B. x = 3; y = 6 C. x = 7; y = 9 D. x = 10; y = 4 E. x = 10; y = 7 The sum n consecutive integers is give by: \(Sum=\frac{(2a_1+n1)*n}{2}\) (check Number Theory chapter of Math Book for more: mathnumbertheory88376.html); Notice that: If \(n=even=2*odd\), so when \(n\) (# of consecutive integers) is even but not a multiple 4 then \(Sum=\frac{(2a_1+n1)*n}{2}=\frac{(even+evenodd)*(2*odd)}{2}=odd*odd=odd\); If \(n=even=2*even\), so when \(n\) is a multiple 4 then \(Sum=\frac{(2a_1+n1)*n}{2}=\frac{(even+evenodd)*(2*even)}{2}=odd*even=even\); That's because a set of even number of consecutive integers has half even and half odd terms. The sum of even terms is obviously even. As for odd terms: their sum is even if their number is even (so total # of terms is multiple of 4) and their sum is odd if their number is odd (so total number of terms is even but not a multiple of 4); So, the sum of 10 (not a multiple of 4) consecutive integers will be odd (the sum of 5 even and 5 odd integers) and the sum of 4 (multiple of 4) consecutive integers will be even (the sum of 2 even and 2 odd integers), so option D is not possible. Answer: D. Hope it's clear. Brunel , Sum = n/2( 2a +(n1)d ).....i still not able to get ur sum formula . I guess for consecutive number you are taking d=1 ...



Math Expert
Joined: 02 Sep 2009
Posts: 65761

Re: sum of x consecutive positive integers
[#permalink]
Show Tags
01 Dec 2013, 05:20
archit wrote: Bunuel wrote: If a is the sum of x consecutive positive integers. b is the sum of y consecutive positive integers. For which of the following values of x and y is it impossible that a = b?A. x = 2; y = 6 B. x = 3; y = 6 C. x = 7; y = 9 D. x = 10; y = 4 E. x = 10; y = 7 The sum n consecutive integers is give by: \(Sum=\frac{(2a_1+n1)*n}{2}\) (check Number Theory chapter of Math Book for more: mathnumbertheory88376.html); Notice that: If \(n=even=2*odd\), so when \(n\) (# of consecutive integers) is even but not a multiple 4 then \(Sum=\frac{(2a_1+n1)*n}{2}=\frac{(even+evenodd)*(2*odd)}{2}=odd*odd=odd\); If \(n=even=2*even\), so when \(n\) is a multiple 4 then \(Sum=\frac{(2a_1+n1)*n}{2}=\frac{(even+evenodd)*(2*even)}{2}=odd*even=even\); That's because a set of even number of consecutive integers has half even and half odd terms. The sum of even terms is obviously even. As for odd terms: their sum is even if their number is even (so total # of terms is multiple of 4) and their sum is odd if their number is odd (so total number of terms is even but not a multiple of 4); So, the sum of 10 (not a multiple of 4) consecutive integers will be odd (the sum of 5 even and 5 odd integers) and the sum of 4 (multiple of 4) consecutive integers will be even (the sum of 2 even and 2 odd integers), so option D is not possible. Answer: D. Hope it's clear. Brunel , Sum = n/2( 2a +(n1)d ).....i still not able to get ur sum formula . I guess for consecutive number you are taking d=1 ... Yes, the common difference between the numbers in a set of consecutive numbers is 1.
_________________



Intern
Joined: 09 Feb 2013
Posts: 1

Re: If a is the sum of x consecutive positive integers. b is the
[#permalink]
Show Tags
01 Dec 2013, 07:09
devinawilliam83 wrote: If a is the sum of x consecutive positive integers. b is the sum of y consecutive positive integers. For which of the following values of x and y is it impossible that a = b?
A. x = 2; y = 6 B. x = 3; y = 6 C. x = 7; y = 9 D. x = 10; y = 4 E. x = 10; y = 7 this is a very bad question ... no way a GMAT question... very time consuming ...



Intern
Joined: 21 Dec 2014
Posts: 4

Re: If a is the sum of x consecutive positive integers. b is the
[#permalink]
Show Tags
05 Jan 2015, 18:06
Excelente Private scores, that is a great approache which can allow us to face a gmat question like that!



Manager
Joined: 24 Dec 2014
Posts: 101
Concentration: Strategy, General Management

Re: If a is the sum of x consecutive positive integers. b is the
[#permalink]
Show Tags
05 Jan 2015, 20:03
Although time consuming, my approach is similar to others.
Sum of 2: 1+2 or 2+1 (odd+even or even+odd) = Odd Sum of 3: 1+2+1 or 2+1+2 = Even/Odd Sum of 4: 1+2+1+2 or 2+1+2+1 (Same both way) = Even Etc. etc.
I got the question right over 2 minutes, hope this helps...
Etc. etc.



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 10780
Location: Pune, India

Re: If a is the sum of x consecutive positive integers. b is the
[#permalink]
Show Tags
05 Jan 2015, 23:15
mdcash wrote: devinawilliam83 wrote: If a is the sum of x consecutive positive integers. b is the sum of y consecutive positive integers. For which of the following values of x and y is it impossible that a = b?
A. x = 2; y = 6 B. x = 3; y = 6 C. x = 7; y = 9 D. x = 10; y = 4 E. x = 10; y = 7 this is a very bad question ... no way a GMAT question... very time consuming ... Actually, it isn't that bad. There is the logical approach of number properties that you can use but even if that doesn't work out in the limited time, go with brute force! A. x = 2; y = 6 Sum of first 6 numbers is 6*7/2 = 21 Can the sum of two consecutive numbers be 21? Sure 10 and 11. Out B. x = 3; y = 6 Sum of first 6 numbers is 21. Can sum of three consecutive numbers be 21? Divide 21 by 3 to get 7. The three numbers will be 6, 7, 8. Out C. x = 7; y = 9 Sum of first 9 numbers is 9*10/2 = 45. Can sum of 7 consecutive numbers be 45? 45 is not divisible by 7 so this will not work. Try another method: Sum1 = Sum2 7*Mean1 = 9*Mean2 If Mean1 = 9 and Mean2 = 7, it will satisfy. i.e. 6, 7, 8, 9, 10, 11, 12 and 3, 4, 5, 6, 7, 8, 9, 10, 11 Out D. x = 10; y = 4 10*Mean1 = 4*Mean2 Mean1/Mean2 = 2/5 Both Mean1 and Mean2 must be fractions "something.5" (Even consecutive numbers) Also, Mean1 = 2x and Mean2 = 5x Hard. Hold it. E. x = 10; y = 7 10*Mean1 = 7*Mean2 Mean1/Mean2 = 7/10 Mean1 must be a fraction "something.5" (10 consecutive numbers) and Mean2 must be an integer (7 consecutive numbers) Such as 3.5/5 but you don't have 10 positive integers around 3.5 So perhaps 10.5/15 The numbers are 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 and 12, 13, 14, 15, 16, 17, 18 Out Answer (D) by elimination.
_________________
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
Status: I am ready!
Joined: 05 Nov 2014
Posts: 40
Location: India

Re: If a is the sum of x consecutive positive integers. b is the
[#permalink]
Show Tags
16 Jun 2015, 22:17
devinawilliam83 wrote: If a is the sum of x consecutive positive integers. b is the sum of y consecutive positive integers. For which of the following values of x and y is it impossible that a = b?
A. x = 2; y = 6 B. x = 3; y = 6 C. x = 7; y = 9 D. x = 10; y = 4 E. x = 10; y = 7 Ok, I did it simply in the head...here it is x = 10 : (sum of 5 even no.+ sum of 5 odd no.)=(even+odd)=odd number over all. y = 4 : (sum of 2 even no.+sum of 2 odd no.)=(even+even)=even number over all. Hope this helps!



Intern
Joined: 10 May 2015
Posts: 24

Re: If a is the sum of x consecutive positive integers. b is the
[#permalink]
Show Tags
27 Jun 2015, 07:05
Bunuel wrote: If a is the sum of x consecutive positive integers. b is the sum of y consecutive positive integers. For which of the following values of x and y is it impossible that a = b?A. x = 2; y = 6 B. x = 3; y = 6 C. x = 7; y = 9 D. x = 10; y = 4 E. x = 10; y = 7 The sum n consecutive integers is give by: \(Sum=\frac{(2a_1+n1)*n}{2}\) (check Number Theory chapter of Math Book for more: mathnumbertheory88376.html); Notice that: If \(n=even=2*odd\), so when \(n\) (# of consecutive integers) is even but not a multiple 4 then \(Sum=\frac{(2a_1+n1)*n}{2}=\frac{(even+evenodd)*(2*odd)}{2}=odd*odd=odd\); If \(n=even=2*even\), so when \(n\) is a multiple 4 then \(Sum=\frac{(2a_1+n1)*n}{2}=\frac{(even+evenodd)*(2*even)}{2}=odd*even=even\); That's because a set of even number of consecutive integers has half even and half odd terms. The sum of even terms is obviously even. As for odd terms: their sum is even if their number is even (so total # of terms is multiple of 4) and their sum is odd if their number is odd (so total number of terms is even but not a multiple of 4); So, the sum of 10 (not a multiple of 4) consecutive integers will be odd (the sum of 5 even and 5 odd integers) and the sum of 4 (multiple of 4) consecutive integers will be even (the sum of 2 even and 2 odd integers), so option D is not possible. Answer: D. Hope it's clear. Hi Bunuel, Could you please explain why we're only dealing with n=even here? Also I'm unable to understand the part "As for odd terms: their sum is even if their number is even (so total # of terms is multiple of 4) and their sum is odd if their number is odd (so total number of terms is even but not a multiple of 4);". Can you please clarify? Thanks



Intern
Joined: 03 Jul 2015
Posts: 10
Location: India

Re: If a is the sum of x consecutive positive integers. b is the
[#permalink]
Show Tags
04 Sep 2015, 11:23
devinawilliam83 wrote: Hi, I am struggling with the explanation.Here is what I had done. but was unable to eliminate answers the premise was that for x and y to be equal both should be either even or odd Please take a look and let me know your approach is correct but have made mistake in the case of n=4. If you have 4 terms in the consecutive sequence, sum will always be even \(n/2 *[2*a + (n1)d]\) since n=4 & d = 1, the expression reduces to: \(2 * [2*a + 3]\) which is always even



Intern
Joined: 07 Sep 2015
Posts: 1

Re: If a is the sum of x consecutive positive integers. b is the
[#permalink]
Show Tags
11 Sep 2015, 06:44
Consider the following scenario :
x=7, y=9
6,7,8,9,10,11,12 are 7 consecutive terms with sum (a) =63
3,4,5,6,7,8,9,10,11 are 9 consecutive term with sum (b) = 63
Thus, a=b and option C is correct.



EMPOWERgmat Instructor
Status: GMAT Assassin/CoFounder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 17264
Location: United States (CA)

Re: If a is the sum of x consecutive positive integers. b is the
[#permalink]
Show Tags
27 Jun 2018, 14:47
Hi All, We're told that A is the sum of X CONSECUTIVE positive integers and B is the sum of Y CONSECUTIVE positive integers. We're asked for which of the following values of X and Y is it IMPOSSIBLE that A = B. This question is based on a subtle Number Property rule (and if you know the rule, then the question becomes a whole lot easier). You can certainly TEST THE ANSWERS to prove what's possible and what's not: Answer A: If X = 2 and Y = 6 you could have A = 10 + 11 = 21 B = 1 + 2 + 3 + 4 + 5 + 6 = 21 In this example, A EQUALS B, so this is not the answer we're looking for. Answer B: If X = 3 and Y = 6 you could have A = 6 + 7 + 8 = 21 B = 1 + 2 + 3 + 4 + 5 + 6 = 21 In this example, A EQUALS B, so this is not the answer we're looking for. You could continue with this method to figure out which of the other 3 answers are possible and which is not. With the following Number Property though, you'll be able to quickly solve this problem: Take a look at answer D.... X = 10 and Y = 4 The sum of 10 consecutive positive integers will ALWAYS be ODD... since there are 5 ODD numbers (try it and you'll see). The sum of 4 consecutive positive integers will ALWAYS be EVEN... since there 2 ODD numbers (again, try it and you'll see) So, A can NEVER EQUAL B in this circumstance. Final Answer: GMAT assassins aren't born, they're made, Rich
_________________
Contact Rich at: Rich.C@empowergmat.comThe Course Used By GMAT Club Moderators To Earn 750+ souvik101990 Score: 760 Q50 V42 ★★★★★ ENGRTOMBA2018 Score: 750 Q49 V44 ★★★★★



Oxford Saïd School Moderator
Joined: 17 Jun 2019
Posts: 47
Location: United Kingdom
GMAT 1: 750 Q48 V45
WE: Information Technology (Insurance)

Re: If a is the sum of x consecutive positive integers. b is the
[#permalink]
Show Tags
05 Jul 2019, 09:44
PerfectScores wrote: devinawilliam83 wrote: If a is the sum of x consecutive positive integers. b is the sum of y consecutive positive integers. For which of the following values of x and y is it impossible that a = b?
A. x = 2; y = 6 B. x = 3; y = 6 C. x = 7; y = 9 D. x = 10; y = 4 E. x = 10; y = 7 Sum of 2 consecutive integers = Odd Sum of 4 consecutive integers = Even Sum of 6 consecutive integers = Odd Sum of 8 consecutive integer = Even and so on is the pattern. If we look at answer option D. Sum of 10 consecutive integer = Odd, Sum of 4 consecutive integers = Even. Odd cannot be equal to Even. Hence the answer is D. How did you eliminate option B with this method? Sum of 3 consecutive integers is even and 6 consecutive integers is odd?
_________________




Re: If a is the sum of x consecutive positive integers. b is the
[#permalink]
05 Jul 2019, 09:44



Go to page
1 2
Next
[ 22 posts ]

