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Thread: Showing that a and b are relatively prime

  1. #1
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    Showing that a and b are relatively prime

    Back again with more "optional" homework!

    Question: "Let a and b be positive integers. Suppose there exist integers j and k such that aj + bk = 1. Show that a and b are relatively prime."

    All I have so far is that a and b are relatively prime iff the gcd(a,b) = 1. Let a and b be elements of all positive integers.
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  2. #2
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    Re: Showing that a and b are relatively prime

    Use "proof by contradiction"- If a and b are not relatively prime. Then there exist positive integers, p, c, and d such that a= pc and b= pd. So aj+ bk= pcj+ pdk= p(cj+ dk)= 1.
    Last edited by HallsofIvy; Mar 6th 2017 at 02:58 PM.
    Thanks from azollner95
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    Re: Showing that a and b are relatively prime

    Thank you so much! So would that be the conclusion of the problem, or do I need a finalizing statement, something like "because the statement/work above is not true thus, a and b are relatively prime"?
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    Re: Showing that a and b are relatively prime

    You might want to say why the statement is true!
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    Re: Showing that a and b are relatively prime

    And that is just thing, I simply don't know WHY the statement is true!
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    Re: Showing that a and b are relatively prime

    So would I say that because the information (that you supplied) is equal to 1, and two numbers are relatively prime if they are equal to 1?
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