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# Which of the following cannot be the sum of two or more cons

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Which of the following cannot be the sum of two or more cons [#permalink]  25 Mar 2013, 09:00
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Which of the following cannot be the sum of two or more consecutive positive integers?

(A) 3^7
(B) 4^6
(C) 5^5
(D) 6^4
(E) 7^3

This Week's Challenge Problem from MGMAT : "Consecutive Positive Madness"

https://www.manhattangmat.com/challenge_thisweek.cfm?submitted=1

I got this wrong ... Can anybody please explain?
[Reveal] Spoiler: OA

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Last edited by Bunuel on 26 Mar 2013, 01:14, edited 1 time in total.
Renamed the topic and edited the question.
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Re: Challenge Problem from MGMAT: Consecutive Positive Madness [#permalink]  25 Mar 2013, 09:32
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KUDOS
Every natural number not of the form 2^k for some natural number k can be written as the sum of two or more consecutive positive integers.
I solved this question manually but later found on internet that numbers which can be expressed as 2^n, n is a natural number, cannot be represented as sum of 2 or more consecutive numbers.

Please give a kudo if you like my explanation.
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Re: Challenge Problem from MGMAT: Consecutive Positive Madness [#permalink]  25 Mar 2013, 20:45
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TheNona wrote:
This Week's Challenge Problem from MGMAT : "Consecutive Positive Madness"

https://www.manhattangmat.com/challenge_thisweek.cfm?submitted=1

Which of the following cannot be the sum of two or more consecutive positive integers?

(A) 3^7 (B) 4^6 (C) 5^5 (D) 6^4 (E) 7^3

I got this wrong ... Can anybody please explain?

There are lots of properties of numbers and GMAT does not expect you to know them. So a question that appears of GMAT must be solvable without knowing the properties. So think hard about what you know and what you can apply. Use pattern recognition.

Try some numbers to start off:
1+2 = 3
2+3 = 5
3+4 = 7
ok, so is there a pattern here? We are getting all odd numbers. 3, 5, 7, 9, 11 etc. Every odd number can be written as sum of two consecutive numbers. Why? Say an odd number is N. When you divide it by 2, you get half of it which has a .5. You take the integer above it and below it and they will add up to give N
e.g. N = 11.
11/2 = 5.5 so take numbers 5 and 6 and they will add to give 11. Why? because 5.5 is the arithmetic mean of 2 consecutive numbers:5 and 6.

Takeaway: Every odd number can be written as sum of two consecutive integers.

So rule out (A), (C) and (E).

Now, try to sum up three consecutive numbers.
1+2+3 = 6
2+3 +4 = 9 (ignore odd numbers so we have already dealt with them)
3+4+5 = 12
4+5+6 = 15
5+6+7 = 18
6+7+8 = 21
7+8+9 = 24

You are getting all multiples of 3. The important thing is that no multiple of 3 is getting skipped. You are getting all of them. Hence we can represent all multiples of 3 as sum of 3 numbers. Hence (D) is also out since it is a multiple of 3.
Now think why?
Sum of three consecutive integers is given by (n-1) + n + (n + 1) = 3n

Takeaway: Sum of any three consecutive numbers will be a multiple of 3.

Hence answer must be (B) i.e. 4^6
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Verbal Forum Moderator Joined: 10 Oct 2012 Posts: 629 Followers: 66 Kudos [?]: 837 [0], given: 135 Re: Challenge Problem from MGMAT: Consecutive Positive Madness [#permalink] 25 Mar 2013, 21:23 Expert's post TheNona wrote: This Week's Challenge Problem from MGMAT : "Consecutive Positive Madness" https://www.manhattangmat.com/challenge_thisweek.cfm?submitted=1 Which of the following cannot be the sum of two or more consecutive positive integers? (A) 3^7 (B) 4^6 (C) 5^5 (D) 6^4 (E) 7^3 We know that the sum of consecutive integers is :S = n/2(f+l) --> Here n is the number of terms and f and l are the first and last terms respectively. Now, we have been told that n>=2. We can have only the following cases : f = odd, l = odd, n = (odd-odd)+1 = odd. Thus, S = odd*even/2. f = odd, l = even, n = (even-odd)+1 = even. S = odd.even/2. Similarly, for the other two combinations also, the sum S = odd*even/2. Now, out of the given options, all the options can be in this pattern except 4^6 = 2^12. B. _________________ Intern Joined: 16 Jan 2013 Posts: 23 Followers: 0 Kudos [?]: 3 [0], given: 17 Re: Which of the following cannot be the sum of two or more cons [#permalink] 04 Apr 2013, 07:43 TheNona wrote: Which of the following cannot be the sum of two or more consecutive positive integers? (A) 3^7 (B) 4^6 (C) 5^5 (D) 6^4 (E) 7^3 This Week's Challenge Problem from MGMAT : "Consecutive Positive Madness" https://www.manhattangmat.com/challenge_thisweek.cfm?submitted=1 I got this wrong ... Can anybody please explain? Is this a 700 level question ? answer simply depends upon logic of even and odd number Intern Joined: 16 Jan 2013 Posts: 23 Followers: 0 Kudos [?]: 3 [0], given: 17 Re: Which of the following cannot be the sum of two or more cons [#permalink] 04 Apr 2013, 07:44 jbisht wrote: TheNona wrote: Which of the following cannot be the sum of two or more consecutive positive integers? (A) 3^7 (B) 4^6 (C) 5^5 (D) 6^4 (E) 7^3 This Week's Challenge Problem from MGMAT : "Consecutive Positive Madness" https://www.manhattangmat.com/challenge_thisweek.cfm?submitted=1 I got this wrong ... Can anybody please explain? Is this a 700 level question ? answer simply depends upon logic of even and odd number 4^(any Number) = will never be odd Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 6062 Location: Pune, India Followers: 1597 Kudos [?]: 8951 [0], given: 195 Re: Which of the following cannot be the sum of two or more cons [#permalink] 05 Apr 2013, 01:50 Expert's post jbisht wrote: TheNona wrote: Which of the following cannot be the sum of two or more consecutive positive integers? (A) 3^7 (B) 4^6 (C) 5^5 (D) 6^4 (E) 7^3 This Week's Challenge Problem from MGMAT : "Consecutive Positive Madness" https://www.manhattangmat.com/challenge_thisweek.cfm?submitted=1 I got this wrong ... Can anybody please explain? Is this a 700 level question ? answer simply depends upon logic of even and odd number That's the thing about GMAT questions - All 700- 800 level questions will also simply depend on the logic of something simple. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: Which of the following cannot be the sum of two or more cons [#permalink]  21 Jun 2014, 06:30
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Re: Which of the following cannot be the sum of two or more cons [#permalink]  13 Sep 2015, 09:50
Hello from the GMAT Club BumpBot!

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Re: Which of the following cannot be the sum of two or more cons [#permalink]  13 Sep 2015, 17:13
whenever if you sum two consecutive positive integers,
one is odd another one is even, sum always will be odd.
so we can easily take away options A ,C and E beacuse all are odd numbers .
Whenever if we power odd numbers how many times sum is odd.

Now remaining is 4^6and 6^4.
both are even numbers if you power it , its sum always even.

Now if you sum two numbers is odd.if you sum three numbers sum can be even or odd.
But here sum of even power numbers is even.
any sum of three numbers is divisible by 3.

Take sum of 4^6 and 6^4.

first take 6^4 is 36*36.
both numbers are divisible by3.

Now look 4^6. sum is 16*16*16.
all three numbers not divisibly by 3.

So option B is not a sum of consecutive numbers..
Re: Which of the following cannot be the sum of two or more cons   [#permalink] 13 Sep 2015, 17:13
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