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If x is an odd integer, which of the following must be an odd integer?

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If x is an odd integer, which of the following must be an odd integer?  [#permalink]

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New post 16 Oct 2019, 00:24
00:00
A
B
C
D
E

Difficulty:

  15% (low)

Question Stats:

83% (00:43) correct 17% (00:56) wrong based on 42 sessions

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Re: If x is an odd integer, which of the following must be an odd integer?  [#permalink]

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New post 16 Oct 2019, 00:47
Bunuel wrote:
If x is an odd integer, which of the following must be an odd integer?

A. \(x^4 + 1\)

B. \(x^4 − 1\)

C. \((x+1)^5 − 2\)

D. \((x+1)^5 + 1\)

E. \((x+1)^6 + 2\)


A. \(x^4 + 1\) --> Odd^4 + 1 = Odd + 1 = Even

B. \(x^4 − 1\) --> Odd^4 - 1 = Odd - 1 = Even

C. \((x+1)^5 − 2\) --> Even^5 - 2 = Even - 2 = Even

D. \((x+1)^5 + 1\) --> Even^5 + 1 = Even + 1 = Odd

E. \((x+1)^6 + 2\) --> Even^6 + 2 = Even + 2 = Even

IMO Option D
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Re: If x is an odd integer, which of the following must be an odd integer?  [#permalink]

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New post 16 Oct 2019, 08:20
Bunuel wrote:
If x is an odd integer, which of the following must be an odd integer?

A. \(x^4 + 1\)

B. \(x^4 − 1\)

C. \((x+1)^5 − 2\)

D. \((x+1)^5 + 1\)

E. \((x+1)^6 + 2\)


Plug in value and check (Since , solution by algebric/property based method has already been posted above )

Let x = 3

(A) 81 + 1 = 82 (Even)
(B) 81 - 1 = 80 (Even)
(C) 1024 -2 - 1022 (Even)
(D) 1024 + 1 = 1025 (Odd)
(E) 4096 + 2 = 4098 (Even)

Hence, Answer must be (D)

PS : Its easier to work with x = 1
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If x is an odd integer, which of the following must be an odd integer?  [#permalink]

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New post 20 Oct 2019, 12:16

Solution



Given
    • x is an odd integer

To find
    • The option that is always an odd integer

Approach and Working out
Let’s check each option.

Option A: \(x^4\)+1
    • x = Odd => \(x^4\) = odd
    • 1 = Odd
      o \(x^4\)+1 = odd + Odd = Even

Option B: \(x^4\)−1
    • x = Odd => \(x^4\) = odd
    • 1 = Odd
      o \(x^4\)-1 = odd - Odd = Even

Option C: \((x+1)^5\)−2
    • x = Odd
    • 1 = Odd
      o x + 1 = odd + Odd = Even
      o \((x+1)^5\) = Even
    • 2 = Even
      o \((x+1)^5\)−2 = Even – Even = Even

Option D: \((x+1)^5\)+1
    • x = Odd
    • 1 = Odd
      o x + 1 = odd + Odd = Even
      o \((x+1)^5\) = Even
    • 1 = Odd
      o \((x+1)^5\) + 1 = Even + Odd= Odd

Option E: \((x+1)^6\)+2

We don’t need to check this option as option D is already correct.

Thus, option D is the correct answer.

Correct Answer: Option D
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If x is an odd integer, which of the following must be an odd integer?   [#permalink] 20 Oct 2019, 12:16
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