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Math Help - find the remainder

  1. #1
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    find the remainder

    find the remainder when
    2^1990 is divided by 1990..

    plz explain the theorems involved in ur solution..

    thanks
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  2. #2
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    Quote Originally Posted by ramanujam View Post
    find the remainder when
    2^1990 is divided by 1990..

    plz explain the theorems involved in ur solution..

    thanks
    2 \equiv 2 (\bmod 1990)

    2^{11} \equiv 2^{11} = -58 (\bmod 1990)

    2^{22} \equiv (-58)^2 = 3364 \equiv -616 (\bmod 1990)

    2^{44} \equiv (-616)^2 = 379456 \equiv -634 (\bmod 1990 )

    2^{88} \equiv (-634)^2 = -24 (\bmod 1990)

    2^{264} \equiv (-24)^3 = -13824 \equiv 106 (\bmod 1990)

    2^{528} \equiv (106)^2 = 11236 \equiv -704 (\bmod 1990 )

    2^{1056} \equiv (-704)^2 =495616 \equiv 106 (\bmod 1990)

    2^{1848}=2^{1056} \cdot 2^{528} \cdot 2^{264} \equiv 106 \cdot (-704) \cdot 106 \equiv 106 (\bmod 1990 )

    2^{1936} = 2^{1848}\cdot 2^{88}\equiv 106 \cdot (-24) \equiv -554 (\bmod 1990)

    2^{1980} = 2^{1936} \cdot 2^{44} = (-554)\cdot (-634) \equiv 996 (\bmod 1990)

    2^{1990} = 2^{1980}\cdot 2^{10} \equiv 996 \cdot 1024 \equiv 1024 (\mod 1990)
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  3. #3
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    second approach

    Thanks perfect hacker..
    a second approach to the problem would be
    1990=2.5.199
    we use euler's totient function here which states
    for a prime ; this is because only the multiples of are not relatively prime to .
    1990/2=5.199
    hence
    by using euler's totient theorem

    or by using carmichael's theorem
    we get

    then
    hence remainder when 2^1990 divided by 1990 is 1024
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  4. #4
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    Quote Originally Posted by ramanujam View Post
    Thanks perfect hacker..
    a second approach to the problem would be
    1990=2.5.199
    That is a nicer approach. But there is a problem here. That approach is not your own! If you use someone else's idea at least say something like.

    "On a different forum I recieved the following answer ...."

    Than that would be okay, but not the way you did it!
    And furthermore you dishonor Ramanajuan's name with such blashphemy
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  5. #5
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    yes u right..
    the approach is not mine..but i never said i had done it..
    m sorry i forgot to mention where i got the second solution from..
    i just thought i did share it with u..

    i'll keep in mind to share from where and how i got my answer from now on..
    Last edited by ramanujam; May 31st 2007 at 11:17 PM.
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  6. #6
    The Gladiator
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    Red face Please Help me about this step.

    Can anybody helpme out, how this last but one step..i.e.
    2^1989 = 2^1989 - 5.396[1990/2]
    i just can't understand this step...
    @ThePerfectHacker,
    can you explain...??

    Thanks

    Quote Originally Posted by ramanujam View Post
    Thanks perfect hacker..
    a second approach to the problem would be
    1990=2.5.199
    we use euler's totient function here which states
    for a prime ; this is because only the multiples of are not relatively prime to .
    1990/2=5.199
    hence
    by using euler's totient theorem

    or by using carmichael's theorem
    we get

    then
    hence remainder when 2^1990 divided by 1990 is 1024
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