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Trouble setting up to prove unique factorization

  1. Aug 16, 2014 #1
    This lemma the book states, I can't make sense of it.

    Lemma: If a,b[itex]\in[/itex] [itex]Z[/itex] and b > 0, there exist q,r [itex]\in[/itex] [itex]Z[/itex] such that a = qb + r with 0 [itex]\leq[/itex] r < b.

    Proof: Consider the set of all integers of the form a-xb with x [itex]\in[/itex] [itex]Z[/itex]. This set includes positive elements. Let r = a - qb be the least nonnegative element in this set. We claim that 0[itex]\leq[/itex] r < b. If not, r = a - qb [itex]\geq[/itex] b and so 0[itex]\leq[/itex] a-(q+1)b<r, which contradicts the minimality of r.

    Can someone help explain this to me? Why can't a = 3, and b = 8. Then no q,r could exist so that r is less than b, right?
     
  2. jcsd
  3. Aug 16, 2014 #2
    Is this considered a textbook style question? Sorry, how can I move it? I just saw number theory and posted
     
  4. Aug 16, 2014 #3
    What is wrong with [itex] a=3 [/itex] and [itex] b=8[/itex]? Here you simply take [itex] q=0 [/itex] and [itex] r=3 [/itex] (there is no condition that says either q or r have to be nonzero) to get [itex] 3=(0)(8) +3 [/itex] and [itex] 0\leq 3< 8 [/itex] as required.
     
  5. Aug 16, 2014 #4
    Wow I'm quitting now

    thanks
     
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