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Thread: Positive Integers x & y

  1. #1
    Super Member harpazo's Avatar
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    Positive Integers x & y

    If positive integers x and y are NOT both odd, which of the following must be even?

    A. xy
    B. x + y
    C. x - y
    D. x + y - 1
    E. 2(x + y) - 1

    My Effort:

    I decided to experiment by letting x be 3 and y be 2.
    Doing this quickly revealed the fact that choice A and D both
    yield an even number. The book's answer is A.

    Question:

    Why is choice D not the answer?
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    Re: Positive Integers x & y

    Quote Originally Posted by harpazo View Post
    If positive integers x and y are NOT both odd, which of the following must be even?

    A. xy
    B. x + y
    C. x - y
    D. x + y - 1
    E. 2(x + y) - 1

    My Effort:

    I decided to experiment by letting x be 3 and y be 2.
    Doing this quickly revealed the fact that choice A and D both
    yield an even number. The book's answer is A.

    Question:

    Why is choice D not the answer?
    What if $x~\&~y$ are both even then $x+y-1$ is ?
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    Re: Positive Integers x & y

    Quote Originally Posted by harpazo View Post
    Why is choice D not the answer?
    Because the question talks about all possible pairs of integers, not just 2 and 3.
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    Re: Positive Integers x & y

    A correct way to approach the question is, for each point A, B, C, D and E to consider the three permitted cases:
    1. $x$ and $y$ are both even, that is $x=2a$, $y=2b$;
    2. $x$ is even and $y$ is odd, that is $x=2a$, $y=2b-1$; and
    3. $x$ is odd and $y$ is even, that is $x=2a-1$, $y=2b$.

    Results that are even will have a factor of 2.

    You might see that 2. and 3. here are essentially the same, so you only really need to do one of them. And you might be able to get by setting only $x=2a$ and considering different cases for $y$ where necessary. But essentially you will be doing the above with or without some shortcuts.
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    Re: Positive Integers x & y

    Quote Originally Posted by harpazo View Post
    If positive integers x and y are NOT both odd, which of the following must be even?
    A. xy______B. x + y______C. x - y______D. x + y - 1______E. 2(x + y) - 1
    Why is choice D not the answer?
    To harpazo, I cannot understand how this can be so mysterious.
    Learn this:
    1. The sum of two even integers is even
    2. The sum of two odd integers is even.
    3. The sum of an even integer & an odd integer is odd.
    4. If $n$ is an odd integer then $n-1$ is even.
    5. If $n$ is an even integer then $n-1$ is odd.
    If you learn these then practice applying them to this question,
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    Super Member harpazo's Avatar
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    Re: Positive Integers x & y

    Quote Originally Posted by Plato View Post
    What if $x~\&~y$ are both even then $x+y-1$ is ?
    Let x = 6 and y = 4

    x + y - 1

    6 + 4 - 1

    10 - 1 = 9 is an odd number.

    Meaning?
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    Super Member harpazo's Avatar
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    Re: Positive Integers x & y

    Quote Originally Posted by Archie View Post
    Because the question talks about all possible pairs of integers, not just 2 and 3.
    I can see that every word in word problems is important.
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    Super Member harpazo's Avatar
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    Re: Positive Integers x & y

    Quote Originally Posted by Plato View Post
    To harpazo, I cannot understand how this can be so mysterious.
    Learn this:
    1. The sum of two even integers is even
    2. The sum of two odd integers is even.
    3. The sum of an even integer & an odd integer is odd.
    4. If $n$ is an odd integer then $n-1$ is even.
    5. If $n$ is an even integer then $n-1$ is odd.
    If you learn these then practice applying them to this question,
    Good information.
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