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Math Help - Greatest common divisor Proof (very hard!)

  1. #1
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    Greatest common divisor Proof (very hard!)

    I can't figure out how to prove the following:

    Let p \in N be a prime and c \in Z. Show that either p | c or gcd(p,c)=1.

    Any help is appreciated.
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  2. #2
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    Quote Originally Posted by Roam View Post
    Let p \in N be a prime and c \in Z. Show that either p | c or gcd(p,c)=1.
    Suppose that k=\gcd(p,c)>1.
    So k|p~\&~k|c. But p being prime implies k=1\text{ or }k=p.
    What does that tell you?
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  3. #3
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    Quote Originally Posted by Plato View Post
    Suppose that k=\gcd(p,c)>1.
    So k|p~\&~k|c. But p being prime implies k=1\text{ or }k=p.
    What does that tell you?
    Is that meant to be a contradiction to show that the greatest common divisor of (p,c) is 1 and the k cannot be greater?

    But p being prime implies k=1\text{ or }k=p.
    How do we know that p is a divisable to c (so it can't be the gcf(p,c))?
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  4. #4
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    Quote Originally Posted by Roam View Post
    Is that meant to be a contradiction to show that the greatest common divisor of (p,c) is 1 and the k cannot be greater?
    How do we know that p is a divisable to c (so it can't be the gcf(p,c))?
    You need lessons in basic logic.
    The point is that either k=\gcd(p,c)=1 \text{ or } k=\gcd(p,c)>1.
    Now if k=\gcd(p,c)=1 we are done with the proof.
    So suppose that k=\gcd(p,c)>1 then k \not= 1.
    Thus this means that k=p so p|c.
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