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Need help in Apostol Calculus proof

  1. Sep 20, 2012 #1
    let b denote a fixed positive integer. Prove the following statement by induction: for every integer n≥0, there exist nonnegative integers q and r such that n= qb+r, 0≤r<b.



    Can someone help me on how to solve this question? and how does induction works here?
    thank you
     
  2. jcsd
  3. Sep 20, 2012 #2
    Are there any restrictions on q and b? Otherwise the statement is trivially satisfied by q = n, b = 1, r = 0.
     
  4. Sep 20, 2012 #3
    I think you have to prove by using induction
     
  5. Sep 20, 2012 #4

    jbunniii

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    I don't think you get to choose b.
     
  6. Sep 20, 2012 #5

    jbunniii

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    You can easily check that it's true for [itex]n = 0[/itex]. Now suppose it's true for [itex]n[/itex], so there exist [itex]q[/itex] and [itex]r < b[/itex] such that [itex]n = qb + r[/itex]. Now consider [itex]n + 1[/itex]. A reasonable first step would be to add 1 to both sides of the equation above:
    [tex]n + 1 = qb + r + 1[/tex]
    What does this do for you?
     
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