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If n > 2, then the sum, S, of the integers from 1 through n

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If n > 2, then the sum, S, of the integers from 1 through n [#permalink] New post 07 Jun 2012, 03:53
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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


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
[Reveal] Spoiler: OA
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Re: If n > 2, then the sum, S, of the integers [#permalink] New post 07 Jun 2012, 04:41
S does not necessarily have to be even. For example, when n=5, you have:

S = [(5)(5+1)]/2 = 15, an odd number
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Re: If n > 2, then the sum, S, of the integers [#permalink] New post 07 Jun 2012, 04:49
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)

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Re: If n > 2, then the sum, S, of the integers from 1 through n [#permalink] New post 07 Jun 2012, 13:26
Expert's post
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.

Answer: D.
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Re: If n > 2, then the sum, S, of the integers from 1 through n [#permalink] New post 06 Nov 2012, 03:43
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!
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Re: If n > 2, then the sum, S, of the integers from 1 through n [#permalink] New post 06 Nov 2012, 03:56
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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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Re: If n > 2, then the sum, S, of the integers from 1 through n [#permalink] New post 06 Jan 2013, 00:05
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.
Answer: D
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If n > 2, then the sum, [#permalink] New post 06 Jan 2014, 20:36
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.

[Reveal] Spoiler: My doubt
Though i agree that the OA is right but even option B should be correct.
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Re: If n > 2, then the sum, [#permalink] New post 06 Jan 2014, 20:49
Expert's post
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.

[Reveal] 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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Re: If n > 2, then the sum, [#permalink] New post 07 Jan 2014, 01:54
Expert's post
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.

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


Merging similar topics. please refer to the solutions above.
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Re: If n > 2, then the sum, S, of the integers from 1 through n [#permalink] New post 07 Jan 2014, 22:54
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 :-D
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Re: If n > 2, then the sum, S, of the integers from 1 through n [#permalink] New post 16 Feb 2014, 08:43
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.

Answer: D.


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

Bumpinggg
Re: If n > 2, then the sum, S, of the integers from 1 through n   [#permalink] 16 Feb 2014, 08:43
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