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

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Joined: 02 Sep 2009
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If x is an odd integer, which of the following must be an odd integer?  [#permalink]

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16 Oct 2019, 00:24
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Difficulty:

15% (low)

Question Stats:

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

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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. $$(x+1)^6 + 2$$

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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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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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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)

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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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.

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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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