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Divisibility Questions

  1. Oct 18, 2005 #1
    Two questions here. I know the definitions, but cannot formulate a through proof.

    1.a and b are positive integers. If a^3 | (is divisible by) b^2, then a | (is divisible by) b.

    Now, by definition, I know that a^3*k=b^2, for some k. Also, I know that a * j = b for some j. But where do I go from here?

    2.If p^a || (exactly divides) m, then p^ka || (exactly divides) m^k.

    Again, by definition, p^a | (is divisible by) m and p^a+1 is not divisible by m. Also, p^ka | (is divisible by) m^k and p^ka+1 is not divisible by m^k+1.

    This is all I can get. I just do not know where to go from here. Does anyone have any suggestions?? Thank you all, and you are all very smart on this website, if I have never mentioned that before!
     
    Last edited: Oct 18, 2005
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  3. Oct 18, 2005 #2

    EnumaElish

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    a^3 k = b^2
    a (a^2 k) = b b
    a (a^2 k/b) = b
    j = a^2 k/b

    Show j is the "same kind of number" as k. If k is a positive integer then show that so is j.
     
    Last edited: Oct 18, 2005
  4. Oct 18, 2005 #3
    Ahhhh, I knew it was something with algebra. Thank you very much. I understand that now. Becuase something multiplied by a must mean that a is divisible by b. Thanks much! If only I can get this second one. :mad:
     
  5. Oct 20, 2005 #4

    Gokul43201

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    It's the other way round : a|b means that a divides b, or b is divisible by a.
     
  6. Oct 20, 2005 #5

    Gokul43201

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    I don't see how this has reduced the difficulty of the problem ... :confused:..or what the OP has understood from it.
     
  7. Oct 21, 2005 #6

    matt grime

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    1. One only ever need consider prime factors, with multiplicty, for a direct proof. Or you could prove it by contradiction.

    2. is easier. p^a exactly divides n is the same as saying n=m*p^a where p does not divide m. The solution should just leap out at you.
     
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