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Intern  Joined: 01 Aug 2011
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If n > 2, then the sum, S, of the integers from 1 through n can be cal  [#permalink]

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Difficulty:   35% (medium)

Question Stats: 69% (01:41) correct 31% (01:19) wrong based on 146 sessions

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If n > 2, then the sum, S, of the integers from 1 through n can be calculated by the following formula: S = n(n + 1)/2. Which one of the following statements about S must be true?

A. S is always odd.
B. S is always even.
C. S must be a prime number
D. S must not be a prime number
E. S must be a perfect square

Source: Nova GMAT
Difficulty Level: 600

Originally posted by shreya717 on 07 Jun 2012, 04:53.
Last edited by SajjadAhmad on 05 Jul 2019, 03:54, edited 1 time in total.
Manager  Joined: 19 Mar 2012
Posts: 143
Location: United States
Concentration: Finance, General Management
GMAT 1: 750 Q50 V42 GPA: 3.69
WE: Analyst (Mutual Funds and Brokerage)
Re: If n > 2, then the sum, S, of the integers from 1 through n can be cal  [#permalink]

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1
1
S does not necessarily have to be even. For example, when n=5, you have:

S = [(5)(5+1)]/2 = 15, an odd number
Current Student B
Joined: 29 Mar 2012
Posts: 295
Location: India
GMAT 1: 640 Q50 V26 GMAT 2: 660 Q50 V28 GMAT 3: 730 Q50 V38 Re: If n > 2, then the sum, S, of the integers from 1 through n can be cal  [#permalink]

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Hi,

S = n(n + 1)/2, n > 2
for ex, n = 3, S = 6
n=5, S = 15 = 3*5 and so on

Now,
if n is even,
then, n = 2k, S = k(2k+1)....(a)

if n is odd,
then, n = 2k+1, S = (2k+1)(k+1)...(b)

Depending on value of k (integer), S can be odd/even
But for n>2, it will always be a product of two numbers....(from (a) & (b))

Thus, answer is (D)

Regards,
Math Expert V
Joined: 02 Sep 2009
Posts: 58381
Re: If n > 2, then the sum, S, of the integers from 1 through n can be cal  [#permalink]

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If n > 2, then the sum, S, of the integers from 1 through n can be calculated by the following formula: S = n(n + 1)/2. Which one of the following statements about S must be true?
A. S is always odd.
B. S is always even.
C. S must be a prime number
D. S must not be a prime number
E. S must be a perfect square

Notice that we are asked "which of the following MUST be true, not COULD be true. For such kind of questions if you can prove that a statement is NOT true for one particular set of numbers, it will mean that this statement is not always true and hence not a correct answer.

A. S is always odd --> not necessarily true if n=3 then 1+2+3=6=even.
B. S is always even --> not necessarily true if n=5 then 1+2+3+4+5=15=odd.
C. S must be a prime number --> not true if n=3 then 1+2+3=6=not prime.
E. S must be a perfect square --> not necessarily true if n=3 then 1+2+3=6=not a perfect square.

Only choice D is left.

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Intern  Joined: 02 Nov 2012
Posts: 27
Re: If n > 2, then the sum, S, of the integers from 1 through n can be cal  [#permalink]

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Correct me if I am not right, but since n > 2, S is always even since odd * even = even and 2 is the only even prime number S can never be a prime number!
Math Expert V
Joined: 02 Sep 2009
Posts: 58381
Re: If n > 2, then the sum, S, of the integers from 1 through n can be cal  [#permalink]

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KevinBrink wrote:
Correct me if I am not right, but since n > 2, S is always even since odd * even = even and 2 is the only even prime number S can never be a prime number!

No, that's not correct. If S is always even, then B must also be correct. But if n=5 then 1+2+3+4+5=15=odd.
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Intern  Joined: 12 Jan 2012
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GMAT 1: 720 Q49 V39 Re: If n > 2, then the sum, S, of the integers from 1 through n can be cal  [#permalink]

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shreya717 wrote:
If n > 2, then the sum, S, of the integers from 1 through n can be calculated by the following formula: S = n(n + 1)/2. Which one of the following statements about S must be true?

A. S is always odd.
B. S is always even.
C. S must be a prime number
D. S must not be a prime number
E. S must be a perfect square

Can you please explain between B & D. Both need to be correct in order for the question to be valid right ?

(S needs to be even to be divisible by 2 & S shouldn't be a prime number)

Thanks,
Shreya

S = [n(n+1)]/2 for n>2 S should be divisible by either n or n+1 (for n = odd S is divisible by n and for n=even S is divisible by n+1) so it cannot be a prime no.
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Location: Pune, India
Re: If n > 2, then the sum, S, of the integers from 1 through n can be cal  [#permalink]

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akankshasoneja wrote:
If n > 2, then the sum, S, of the integers from 1 through n can be calculated by the following
formula: S = n(n + 1)/2. Which one of the following statements about S must be true?
(A) S is always odd.
(B) S is always even.
(C) S must be a prime number.
(D) S must not be a prime number.
(E) S must be a perfect square.

Spoiler: :: My doubt
Though i agree that the OA is right but even option B should be correct.

Put n = 5

S = 5*6/2 = 15
S is not always even. It may be even, it may be odd. If the even integer (out of n and n+1) is not a multiple of 4, then S will be odd.
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Intern  Joined: 10 Oct 2013
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GMAT 1: 730 Q50 V38 Re: If n > 2, then the sum, S, of the integers from 1 through n can be cal  [#permalink]

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S= n (n+1) /2 ,
Either n or n+1 , is even & also n > 2,
Thus after dividing by 2, S can be shown to be a product of two distinct numbers (not including 1) ----> S can never be prime . So D it is SVP  Joined: 06 Sep 2013
Posts: 1571
Concentration: Finance
Re: If n > 2, then the sum, S, of the integers from 1 through n can be cal  [#permalink]

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Bunuel wrote:
If n > 2, then the sum, S, of the integers from 1 through n can be calculated by the following formula: S = n(n + 1)/2. Which one of the following statements about S must be true?
A. S is always odd.
B. S is always even.
C. S must be a prime number
D. S must not be a prime number
E. S must be a perfect square

Notice that we are asked "which of the following MUST be true, not COULD be true. For such kind of questions if you can prove that a statement is NOT true for one particular set of numbers, it will mean that this statement is not always true and hence not a correct answer.

A. S is always odd --> not necessarily true if n=3 then 1+2+3=6=even.
B. S is always even --> not necessarily true if n=5 then 1+2+3+4+5=15=odd.
C. S must be a prime number --> not true if n=3 then 1+2+3=6=not prime.
E. S must be a perfect square --> not necessarily true if n=3 then 1+2+3=6=not a perfect square.

Only choice D is left.

Does anyone know why can't the sum be a prime number?

So I began trying to understand this. First since all prime numbers greater than 3 are of the form 6k+1 or 6k-1
Now then let's take 1+6k, that means that 2+3+4......+n cannot be a multiple of 6, but i'm trying to figure out why this can't be true?

Thanks
Cheers
J

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Re: If n > 2, then the sum, S, of the integers from 1 through n can be cal  [#permalink]

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_________________ Re: If n > 2, then the sum, S, of the integers from 1 through n can be cal   [#permalink] 30 Oct 2018, 02:13
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