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A problem with Euler theorem

  1. Apr 17, 2009 #1
    1. The problem statement, all variables and given/known data

    i need to obtain the remainder of the divison [tex] 2^{1000005}[/tex] divided by 55

    2. Relevant equations

    Euler theorem [tex] 2^{\phi (55)}=1 mod(55) [/tex]



    3. The attempt at a solution

    my problem is that applying Euler theorem i reach to the conclusion that the remainder is the same as the value 'a' inside the congruence equation

    [tex] 2^{5}=a mod(55)[/tex] but it would give me that a is negative ¡¡

    it gives me a=-23 or similar using congruences or a =32
     
  2. jcsd
  3. Apr 17, 2009 #2
    phi(55) = 4*10=40

    So, 2^40 = 1

    1000005 = 1000000 + 5 which Modulo 40 is 5

    So 2^2000005 = 2^5 = 32.
     
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