if p|q and p|r, when q>r then ?

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  • 2 months ago

    p|q so q = pa for some a.

    p|r so r = pb for some b.

    Create a counterexample to show the dark blue choice is false.

    Let p = 2, r = 4, and q = 6.

    2|6 and 2|4, but r|q = 4|6 is false.

    For the light blue choice,

    q - r =

    pa - pb =

    p(a - b)

    Since a and b are integers, p|(q - r).

    For the yellow choice 

    qr = papb = p(apb).  Since a, b, and p are integers, p|qr.

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  • Pope
    Lv 7
    2 months ago

    These are all positive integers, right?

    p | q → q = pa, for some positive integer a

    p | r → r = pb, for some positive integer b

    q > r

    pa > pb

    a > b

    a - b > 0

    That makes (a - b) a positive integer.

    q - r = pa - pb

    q - r = p(a - b)

    p | (q - r)

    qr = par

    p | (qr)

    q = pa

    q = pb(a/b)

    q = r(a/b)

    Therefore, r | q only if b | a.

    It cannot be concluded that r | q.

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    • Pope
      Lv 7
      2 months agoReport

      You mean which two are correct. Those would be the two I gave you.

      p | (q - r)
      p | (qr)

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