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Proof Question

  1. Jun 8, 2012 #1
    Prove or disprove:

    There exists an integer "a" such that [itex]ab\equiv\,0\,(mod 3)[/itex] for every integer "b".

    I know I can rewrite the above as [itex]ab=3k[/itex] for some k[itex]\,\in\,\mathbb{Z}[/itex], but other than that I'm not sure where to go. I realize that dividing any of the above will not necessarily result in an integer which contradicts the initial statement, but I'm sorta lost on the wording. Am I on the right path?

    Thanks.
     
  2. jcsd
  3. Jun 8, 2012 #2
    Try dividing both sides by 3 (so the right side is an integer). Do you see an obvious choice for a so that the left side is an integer or are there no choices?
     
  4. Jun 8, 2012 #3
    As long as a is a multiple of 3 it would work. I just don't know how to word that correctly.
     
  5. Jun 8, 2012 #4
    Here's how you might start the proof:

    Indeed, let a=3, then for any b...
     
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