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# If x is an integer, then which of the following statements about x

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Senior Manager
Status: Gathering chakra
Joined: 05 Feb 2018
Posts: 434
If x is an integer, then which of the following statements about x  [#permalink]

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07 Feb 2019, 07:56
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Difficulty:

5% (low)

Question Stats:

84% (01:17) correct 16% (01:49) wrong based on 43 sessions

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If x is an integer, then which of the following statements about x^2 – x – 1 is true?

A) It is always odd.
B) It is always even.
C) It is always positive.
D) It is even when x is even and odd when x is odd.
E) It is even when x is odd and odd when x is even.

Solution:
A) If x is an Odd#, then x^2 is O#*O# so also O#, then - O# we get an Even#, then -1 (O#) answer becomes an O#
x=3, 9-3-1 = 5

B) If x is an Even#, then x^2 is E#, E#-E# = E#, then -1 (O#) we get an O# again.
x=2, 4-2-1 = 1
So there's no way that the result can be even. B is out.

C) When there's exponents and fractions together this issue always comes up.
x=1/2, 1/4 - 1/2 - 1 = -5/4 so not necessarily positive
Same with x=-1/2, 1/4 + 1/2 - 1 = -1/4

D) and E) restate A) and B) but are only half correct, can check using the same examples

Official explanation:
Because x is an integer, x must be either even or odd. If x is even, then x^2 – x must also be even, and therefore x^2 – x – 1 is always odd. If x is odd, then x^2 – x must be even, and again, x^2 – x – 1 is always odd. You can also solve this problem by Plugging In. After Plugging In several values for x and calculating x^2 – x – 1, you will discover that the result is always odd. The correct answer is (A).
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Joined: 02 Sep 2009
Posts: 58398
Re: If x is an integer, then which of the following statements about x  [#permalink]

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07 Feb 2019, 07:58
1
energetics wrote:
If x is an integer, then which of the following statements about x^2 – x – 1 is true?

A) It is always odd.
B) It is always even.
C) It is always positive.
D) It is even when x is even and odd when x is odd.
E) It is even when x is odd and odd when x is even.

Solution:
A) If x is an Odd#, then x^2 is O#*O# so also O#, then - O# we get an Even#, then -1 (O#) answer becomes an O#
x=3, 9-3-1 = 5

B) If x is an Even#, then x^2 is E#, E#-E# = E#, then -1 (O#) we get an O# again.
x=2, 4-2-1 = 1
So there's no way that the result can be even. B is out.

C) When there's exponents and fractions together this issue always comes up.
x=1/2, 1/4 - 1/2 - 1 = -5/4 so not necessarily positive
Same with x=-1/2, 1/4 + 1/2 - 1 = -1/4

D) and E) restate A) and B) but are only half correct, can check using the same examples

Official explanation:
Because x is an integer, x must be either even or odd. If x is even, then x^2 – x must also be even, and therefore x^2 – x – 1 is always odd. If x is odd, then x^2 – x must be even, and again, x^2 – x – 1 is always odd. You can also solve this problem by Plugging In. After Plugging In several values for x and calculating x^2 – x – 1, you will discover that the result is always odd. The correct answer is (A).

$$x^2 – x – 1=odd^2-odd-odd=odd-odd-odd=even-odd=odd$$.

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Re: If x is an integer, then which of the following statements about x   [#permalink] 07 Feb 2019, 07:58
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