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Divisibility in the Integers. Intro to Analysis

  1. Jan 31, 2012 #1
    1. The problem statement, all variables and given/known data


    Prove: If a|b and b|c then a|c.

    Assume a, b and c are integers.

    2. Relevant equations

    none

    3. The attempt at a solution

    If a divides b then that means that there is a

    real integer "r" that is ra=b .

    and since we assume b divides c then c=bs.

    After here I got stuck. I was thinking maybe subsitute b and c for ar and bs, but it doesnt seem to get me anywhere. Thanks in advance.
     
  2. jcsd
  3. Jan 31, 2012 #2

    SammyS

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    You have b in terms of a ? ...
     
  4. Jan 31, 2012 #3
    Well I was thinking like we had 3|15. So a would be 3 and b would be 15. The r would be 5, so ra=b= 5*3=15. Or am I thinking about this wrong?
     
  5. Jan 31, 2012 #4
    If a|b, then there exists an int r such that a*r = b.

    If b|c, then there exists an int s such that s*b = c.

    Since b = a*r, we have s*(a*r)=a*(r*s) = c.

    Since r*s is an integer, this shows that c equals a multiplied by an integer, meaning a|c.
     
  6. Jan 31, 2012 #5

    SammyS

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    If this is a concrete example, you'll need another number.

    You have 3|15, that's like a|b. Now you need b|c, so in the example, 15| ? .
     
  7. Jan 31, 2012 #6
    Oh my goodness....First class on proofs its just simple subistution. Thanks man, cleared up alot!
     
  8. Jan 31, 2012 #7

    SammyS

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    Well, I saw that you were almost there in your original post. I'd rather lead you to discover what's missing than to simply provide the bridge.

    Good luck with the proofs. It can be challenging, coming up with some of them, but very rewarding once you do !
     
  9. Jan 31, 2012 #8
    What sucks about this course is theres only like 10 problems per section. He collects all 10 or so problems, so theres no extra problems to work out. Its not overly difficult, its just a bit different than the math im used too. Computational math is much different than proof based math. Lol.
     
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