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How does this imply this (number theory)

  1. Sep 9, 2010 #1
    so I have
    [tex]2^{1990}=(199k+2)^{10}[/tex]
    expanding I have.
    [tex]2^{1990}=2^{10}+10.2^9. (199k)+\frac{10.9}{1.2} 2^8.(199k)^2+...+10.2. (199k)^9+(199K)^{10}[/tex]-(1)

    now its clear [tex]199|2^{1990}-2^{10}[/tex] since I can take 199 out of the RHS.

    but the book seems to imply that the above equation(1) says [tex]10|2^{1990}-2^{10}[/tex] , but how?I cant see how the equation above says the [tex]10|2^{1990}-2^{10}[/tex] is true..

    Thanks.
     
  2. jcsd
  3. Sep 9, 2010 #2

    jgens

    User Avatar
    Gold Member

    Clearly, 21990-210 = 210(21980-1). Therefore, to show that this number is divisible by 10, it suffices to show that 21980-1 is divisible by 5. You can prove this fact by showing that 5|24n-1 (use induction).
     
  4. Sep 9, 2010 #3
    thanks:)
     
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