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Help with prime factorization proof

  1. Feb 29, 2004 #1
    I have to prove that if ab is divisible by the prime p, and a is not divisible by p, then b is divisible by p.

    In order to prove this, I have to show (a,p)=1. I am not sure what this statement means.

    Then I am supposed to use the fact that 1=sa + tp when s,t are elements of the set of integers. (This statement was already proved in class). Then I figured to multiply across by b so that we get

    b= sab + tpb. I am not sure where to from here. I have not seen to many proofs regarding prime factorization. Thanks

  2. jcsd
  3. Feb 29, 2004 #2


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    This means "The greatest common divisor of a and p is 1". You may have sometimes seen this written as gcd(a, p) = 1.

    Well, you want to know if p divides the LHS of this, and the LHS is equal to the RHS...
  4. Mar 1, 2004 #3
    If ab has a factor p and a don't, then b has the factor. That's logic.

    If a = c + id and b = e - id, it's a bit harder.
    Last edited: Mar 1, 2004
  5. Mar 1, 2004 #4

    matt grime

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    Every result in maths is 'just logic', surely.

    To show there is some content, consider Z{sqrt(5)]

    2 is prime

    2 divides 4=(sqrt5 - 1)(sqrt 5 +1)

    2 divides neither of the terms on the left as they are both prime too.

    so it important that the division algorithm works in Z. Or was that reference to x+iy some indiction of something in the ring Z?
    Last edited: Mar 1, 2004
  6. Jul 5, 2004 #5
    You are almost finished.

    Since you have already arrived at b=sab +tpb, we know that p divides tpb, and p divides sab so that p divides b.

    If there seems a need here for steps, we can look at p(sab/p +tb) =b. Since we know (sab/p +tb) is an integer, we see that b contains the factor p.
    Last edited: Jul 6, 2004
  7. Jul 6, 2004 #6
    Do you enjoy necromancing threads that are months old or something? :P
  8. Aug 4, 2004 #7
    Perhaps it's not true?
  9. Aug 5, 2004 #8
    I hoped I wasn't doing any harm. As for as good, well, I don't know. I thought it added for completeness.
  10. Aug 6, 2004 #9
    Oh no, I was just kidding around when I said that.
  11. Aug 6, 2004 #10


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    Perhaps WHAT'S not true?
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