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If x, y, z are all integers, is xyz odd?

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Bunuel
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If x, y, z are all integers, is xyz odd? [#permalink] New post   07 Feb 2020, 04:28
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If x, y, z are all integers, is xyz odd?

(1) x - 1 is even
(2) (y - z)(x - z) is odd
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B
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If x, y, z are all integers, is xyz odd? [#permalink] New post   07 Feb 2020, 04:38
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If x, y, z are all integers, is xyz odd?
xyz will be odd only if each of x, y, z are odd

(1) x - 1 is even
it means x = odd, but no info about y or z.
NOT Sufficient.

(2) (y - z)(x - z) is odd
it means y-z = odd and x-z = odd
so, if y is odd, z is even : xyz = even
if y is even, z is odd: xyz = even

Sufficient.
Answer = B

if x+/-y = Even, it means both x and y are either odd or even.
If x+/-y = odd, it means one of them is odd and the other one is even
if there are integers a, b, c, d, e... the product will be even if at least one of them is even and the product is odd only if all of them are odd


Bunuel wrote:
If x, y, z are all integers, is xyz odd?

(1) x - 1 is even
(2) (y - z)(x - z) is odd
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Re: If x, y, z are all integers, is xyz odd? [#permalink] New post   07 Feb 2020, 04:39
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If x, y, z are all integers, is xyz odd?

(1) x - 1 is even
(2) (y - z)(x - z) is odd

Statement 1 gives x is odd but don't know if y and z are odd/even - Not Sufficient

Statement 2 gives
(y-z) is odd -> either of them is even i.e. if z=odd, y=even; if z=even, y=odd; z=0, y=odd; y=0, z=odd
and
(x-z) is odd -> either of them is even i.e. if z=odd, x=even; if z=even, x=odd; z=0, x=odd; x=0, z=odd

In anycase, xyz will be even - Therefore sufficient to answer -> B
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Re: If x, y, z are all integers, is xyz odd? [#permalink] New post   07 Feb 2020, 05:16
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If x, y, z are all integers, is xyz odd?

(1) x - 1 is even
(2) (y - z)(x - z) is odd

Solution


Step 1: Analyse Question Stem


    • x, y and z are integers.
    • We need to find if xyz is odd.
      o All x, y, z must be odd for the product xyz to be odd.
      o If at least one of x, y and z is even, xyz will be even.
         So, we need to figure out if all of them i.e. x, y and z are odd or not.
         Or, if at least one of them is even.

Step 2: Analyse Statements Independently


Statement 1: x – 1 is even
    • According to this statement: x is odd.
However, we do not know the nature of y and z .
Hence, statement 1 is not sufficient and we can eliminate answer options A and D
Statement 2: (y-z) (x-z) is odd
    • According to this statement (y-z) (x-z) is odd.
      o (y-z) and (x-z) both are odd.
    • Since (y-z) is odd, either y or z must be even.
      o Hence, xyz = even
Note: We can also consider (x-z) is odd and we will get the same answer.
Thus, the correct answer is Option B.
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Re: If x, y, z are all integers, is xyz odd? [#permalink] New post   23 Feb 2021, 01:34
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Re: If x, y, z are all integers, is xyz odd? [#permalink]
23 Feb 2021, 01:34
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Bunuel
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