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Thread: Show that a and a+2 are relatively prime if and only if a is odd.

  1. #1
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    Thumbs down Show that a and a+2 are relatively prime if and only if a is odd.

    "Let a be a positive integer. Show that a and a+2 are relatively prime if and only if a is odd"

    I don't know how to show if a is odd, nor do I know how to show neither a nor a+2 is relatively prime. This isn't a good start to understanding this optional homework.
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    Re: Show that a and a+2 are relatively prime if and only if a is odd.

    I feel like the first step, based off of the "if and only if" statement is that you have to show how a is odd. I believe showing something is even implies that you take that number by 2n (some variable n), and for odd values you show that a would be multiplied by some number n+1 or 2n+1. I am still very unclear on how to go about this.
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    Re: Show that a and a+2 are relatively prime if and only if a is odd.

    This is the only if direction

    suppose $a$ and $a+2$ are relatively prime and that $a$ is even

    let $a = 2k$

    $a + 2 = 2k+2 = 2(k+1)$

    so $a$ and $a+2$ share the factor $2$ and thus are not relatively prime, so $a$ must be odd

    Now suppose that $a$ is odd

    $a = 2k+1$

    $a+2 = 2(k+1) + 1$

    $2$ is not a factor of either of these as they are both odd so $3$ must be the smallest common factor if one exists.

    Suppose $p\geq 3$ is a factor of $a$. Then the next multiple of $p$ is $a+p > a+2$ so it can't be that $a$ and $a+2$ share $p$ as a factor

    Thus it must be that $a$ and $a+2$ have no common factors and are thus relatively prime.
    Thanks from azollner95
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