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If n is a positive integer, which of the following can’t be the value

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If n is a positive integer, which of the following can’t be the value  [#permalink]

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New post 14 Aug 2018, 17:48
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[Math Revolution GMAT math practice question]

If \(n\) is a positive integer, which of the following can’t be the value of \((n+1)^4-n^4\)?

A. 2465
B. 4641
C. 6096
D. 7825
E. 9855

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Re: If n is a positive integer, which of the following can’t be the value  [#permalink]

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New post 14 Aug 2018, 18:14
MathRevolution wrote:
[Math Revolution GMAT math practice question]

If \(n\) is a positive integer, which of the following can’t be the value of \((n+1)^4-n^4\)?

A. 2465
B. 4641
C. 6096
D. 7825
E. 9855


\((n+1)^4-n^4\) is the difference of some power of consecutive numbers, so it will always be ODD..
O-E=O
E-O=0

C
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Re: If n is a positive integer, which of the following can’t be the value  [#permalink]

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New post 16 Aug 2018, 08:54
=>

If\(n + 1\) is an even number, then\(n\) is an odd number and \((n+1)^4-n^4\)must be an odd number.
If \(n + 1\) is an odd number, then \(n\) is an even number and \((n+1)^4-n^4\)must be an odd number.
All answer choices except for C) are odd numbers.

Therefore, the answer is C.

Answer: C
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Re: If n is a positive integer, which of the following can’t be the value  [#permalink]

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New post 16 Aug 2018, 23:36
Difference between Odd & Even or Even & Odd is always odd. So the answer must be odd. C is the only even number in the given answer choices.

Answer: C.
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Re: If n is a positive integer, which of the following can’t be the value  [#permalink]

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New post 17 Aug 2018, 01:55

Solution



Given:
    • n is a positive integer.

To find:
    • The option which cannot be the value of \((n+1)^4−n^4\)

Approach and Working:
    • Now, \((n+1)^4−n^4\) is subtraction between two consecutive numbers, one of which always be odd and another will always be even.
      o And, O-E or E-O is always Odd.
      o Hence. 6096 cannot be the value of \((n+1)^4−n^4\).

Hence, the correct answer is option C.

Answer: C
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Re: If n is a positive integer, which of the following can’t be the value  [#permalink]

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New post 21 Aug 2018, 23:09
First statement, says that n is a positive integer ie n> 0. Now, they are asking the value of (n+1)^4−n^4. Lets say n is even, so (n+1)^4 – n^4 = Odd – Even = Odd. And if we take n as odd, the expression (n+1)^4−n^4 becomes Even – Odd = Odd. So in both the cases the answer is odd and if we analyse the question it says which of the following cant be the value, hence even cant be the value. So the answer option is C.
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Re: If n is a positive integer, which of the following can’t be the value  [#permalink]

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New post 24 Aug 2018, 06:50
Top Contributor
MathRevolution wrote:
[Math Revolution GMAT math practice question]

If \(n\) is a positive integer, which of the following can’t be the value of \((n+1)^4-n^4\)?

A. 2465
B. 4641
C. 6096
D. 7825
E. 9855


KEY CONCEPTS:
#1) If n is an integer, then n and n+1 are consecutive integers
#2) If n and n+1 are consecutive integers, then one value is ODD and the other value is EVEN
#3) ODD^4 = (ODD)(ODD)(ODD)(ODD) = ODD
#4) EVEN^4 = (EVEN)(EVEN)(EVEN)(EVEN) = EVEN
#5) ODD - ODD = EVEN
#6) ODD - EVEN = ODD
#7) EVEN - ODD = ODD
#8) EVEN - EVEN = EVEN


From #2, there are two possible cases to consider:
case 1: n is EVEN and n+1 is ODD
case 2: n is ODD and n+1 is EVEN

case 1: n is EVEN and n+1 is ODD
In this case, (n+1)^4 - n^4 = ODD^4 - EVEN^4
= ODD - EVEN (from #3 and #4)
= ODD (from #6)

case 2: n is ODD and n+1 is EVEN
In this case, (n+1)^4 - n^4 = EVEN^4 - ODD^4
= EVEN - ODD (from #3 and #4)
= ODD (from #7)

In both cases, (n+1)^4 - n^4 = some ODD integer.

Check the answer choices........
All of the answer choices are ODD, except for answer choice C

Since (n+1)^4 - n^4 can't be EVEN, the correct answer is C

Cheers,
Brent
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Re: If n is a positive integer, which of the following can’t be the value   [#permalink] 24 Aug 2018, 06:50
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