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Divisibility proof writing

  1. Apr 21, 2014 #1
    The problem statement, all variables and given/known data

    this is the original question
    prove: [itex]\forall[/itex] c [itex]\in[/itex] Z, a≠ 0 and b both [itex]\in[/itex] Z$
    a|b ⇔ c*a|c*b

    Then he corrected himself by saying for problem 1: to show that ca | cb implies a | b ... you must assume c NOT = 0 and invoke "Cancellation Property" of Z.

    This kind of confused me but i think I get what he means

    The attempt at a solution

    so I understand that If you have that ca | cb thats like saying that ac=cbq for some q∈ℤ so, if c≠0 you can just take out those c in the both sides of the expression(because of "Cancellation Property" as he said) and you got left a=bq wich means that a|b

    my problem is how do I translate this into a formal proof if and only if proof.
  2. jcsd
  3. Apr 21, 2014 #2
    ok so this is what I've got
    Prove: ∀c∈Z, c≠0 and b both∈Z a|b⇔ca|cb
    a|b if and only if b=ak for some k∈Z
    if and only if cb=cak for some c∈Z
    if and only if ac|cb

    Is this a valid proof? It seems kind of short and it's lacking the "cancelation property" but I'm not sure I understand how to write it any other way
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