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# If a is an odd integer, which of the following must be an even integer

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Math Expert
Joined: 02 Sep 2009
Posts: 51121
If a is an odd integer, which of the following must be an even integer  [#permalink]

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03 Oct 2016, 22:17
00:00

Difficulty:

75% (hard)

Question Stats:

54% (01:39) correct 46% (01:46) wrong based on 254 sessions

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

A. a^4−a+1
B. (a^4−a)(a+1/a)
C. a^4−a^3+a^2+2a
D. (a^3+a^2+a)^2
E. None of the above.

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

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04 Oct 2016, 00:41
Bunuel wrote:
If a is an odd integer, which of the following must be an even integer?

A. a^4−a+1
B. (a^4−a)(a+1/a)
C. a^4−a^3+a^2+2a
D. (a^3+a^2+a)2
E. None of the above.

Let a be 1..

A. 1-1+1..odd
B.(1-1)(2) = 0.. even
C.1-1+1+2..odd
D.1+1+1..odd
E.None

IMO option B.
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If a is an odd integer, which of the following must be an even integer  [#permalink]

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Updated on: 04 Oct 2016, 05:44
Bunuel wrote:
If a is an odd integer, which of the following must be an even integer?

A. a^4−a+1
B. (a^4−a)(a+1/a)
C. a^4−a^3+a^2+2a
D. (a^3+a^2+a)^2
E. None of the above.

IMO B

A. a^4−a+1= odd-odd+odd= even+odd=odd
B. (a^4−a)(a+1/a)= (odd-odd)(even/odd)= even Eg: (5^4-5)(6/5)=even, (3^4-3)(4/3)=even
C. a^4−a^3+a^2+2a= odd-odd+odd+even= even+odd+even= odd
D. (a^3+a^2+a)2= (odd+odd+odd)^2= odd
E. None of the above.[/quote]

Originally posted by Vinayak Shenoy on 04 Oct 2016, 03:02.
Last edited by Vinayak Shenoy on 04 Oct 2016, 05:44, edited 1 time in total.
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Re: If a is an odd integer, which of the following must be an even integer  [#permalink]

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04 Oct 2016, 03:46
Bunuel - can you pls tell if in option D the 2 in the last is for square or product. TIA
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Re: If a is an odd integer, which of the following must be an even integer  [#permalink]

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04 Oct 2016, 05:25
gauravk wrote:
Bunuel - can you pls tell if in option D the 2 in the last is for square or product. TIA

It's squared: $$(a^3+a^2+a)^2$$
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Re: If a is an odd integer, which of the following must be an even integer  [#permalink]

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04 Oct 2016, 05:34
Vinayak Shenoy wrote:
Bunuel wrote:
If a is an odd integer, which of the following must be an even integer?

A. a^4−a+1
B. (a^4−a)(a+1/a)
C. a^4−a^3+a^2+2a
D. (a^3+a^2+a)2
E. None of the above.

IMO D

A. a^4−a+1= odd-odd+odd= even+odd=odd
B. (a^4−a)(a+1/a)= (odd-odd)(even/odd)= even (even/odd) may or may not be even
C. a^4−a^3+a^2+2a= odd-odd+odd+even= even+odd+even= odd
D. (a^3+a^2+a)2= (odd+odd+odd)* even= always even
E. None of the above.
[/quote]

Would D be odd here?
(odd+odd+odd) = odd
(Odd)^even = odd

For eg :- 3^2, 5^2

IMO B i think
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If a is an odd integer, which of the following must be an even integer  [#permalink]

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04 Oct 2016, 05:46
mechky wrote:
Vinayak Shenoy wrote:
Bunuel wrote:
If a is an odd integer, which of the following must be an even integer?

A. a^4−a+1
B. (a^4−a)(a+1/a)
C. a^4−a^3+a^2+2a
D. (a^3+a^2+a)2
E. None of the above.

IMO D

A. a^4−a+1= odd-odd+odd= even+odd=odd
B. (a^4−a)(a+1/a)= (odd-odd)(even/odd)= even (even/odd) may or may not be even
C. a^4−a^3+a^2+2a= odd-odd+odd+even= even+odd+even= odd
D. (a^3+a^2+a)2= (odd+odd+odd)* even= always even
E. None of the above.

Would D be odd here?
(odd+odd+odd) = odd
(Odd)^even = odd

For eg :- 3^2, 5^2

IMO B i think[/quote]

mechky
Kindly note the problem did not show option D was power of 2 initially. When all other powers were mentioned as ^4, ^3 in option D it wasn't mentioned that way.
Also my interpretation was wrong
Corrected it now
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If a is an odd integer, which of the following must be an even integer  [#permalink]

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20 Jan 2017, 06:11
This one is an amazing Question.
I overlooked Option B.
Damn.

Nevertheless here is what i would now -->
Firstly -->Positive exponents does not affect the even/odd nature of any number
Option A-> a^4−a+1=> odd-odd+odd=> even+odd=> odd=> Rejected.
Option B->(a^4−a)(a+1/a)=> (a^3-1)*(a^2+a)=> (odd-odd)*(odd+odd) => even*even => even=> Acceptable.
OptionC->a^4−a^3+a^2+2a=> odd-odd+odd+even=>even+odd=>odd=> Rejected.
Option D->(a^3+a^2+a)^2=>(odd+odd+odd)^2=>(even+odd)^2=>odd^2=> odd=>Rejected.
Option E->None of the above.=> Rejected.

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

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21 Feb 2017, 09:50
I did a substitution.

B was successful with 3 then tried 5 just to be sure and realized this holds true for all odd numbers (multiple of a)(even/a) will always be even.

Just had a question - is there anyways to know if this holds true for all cases i.e. in a scenario where a rule worked for 3 and 5 but would fail for 13..
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Re: If a is an odd integer, which of the following must be an even integer  [#permalink]

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21 Feb 2017, 17:15
Can anyone explain how B satisfies a=3 ??
the answer should be E i believe
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Posts: 7102
Re: If a is an odd integer, which of the following must be an even integer  [#permalink]

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21 Feb 2017, 17:52
2
kparik86 wrote:
Can anyone explain how B satisfies a=3 ??
the answer should be E i believe

Hi,

Firstly lets see the equation..
$$(a^4-a)(a+\frac{1}{a}=(a^3-1)*a*\frac{a^2+1}{a}=(a^3-1)(a^2+1)$$..
Here both a^3-1 and a^2+1 are even thus the answer will be even..

Let's put 3..
$$(3^4-3)(3+\frac{1}{3}=(81-3)*\frac{3^2+1}{3}=(78)(10/3)=26*10=260$$..
Thus EVEN
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Re: If a is an odd integer, which of the following must be an even integer  [#permalink]

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13 Apr 2017, 00:30
Option B

a is an Odd integer. Odd: O, Even: E & Fraction: F

A. a^4−a+1 = O - O + 1 = E + 1 = O
B. (a^4−a)(a+1/a) = (a^4−a)(a^2 + 1)/a = (a^3 - 1)(a^2 + 1) = (O - 1)(O + 1) = E*E = E
C. a^4−a^3+a^2+2a = O - O + O + E = O
D. (a^3+a^2+a)^2 = (O + O + O)^2 = (O)^2 = O
E. None of the above
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Re: If a is an odd integer, which of the following must be an even integer  [#permalink]

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