Results 1 to 5 of 5
Like Tree2Thanks
  • 1 Post By Idea
  • 1 Post By topsquark

Math Help - mod 7

  1. #1
    Member
    Joined
    Nov 2013
    From
    Australia
    Posts
    187
    Thanks
    38

    mod 7

    find the smallest positive integer b which satisfies 3^56=b(mod7)

    I am very new to modular arithmetic and I stubbled upon this question.
    Wouldn't 'b' have a specific solution.
    Asking for the smallest positive integer b confuses me.


    3^{56} - b = 7k  \;\;\text{      Where k is an integer}

    when I put 3^56 into my calculator or into Excel i get a rounded answer. This also surprised me a little.
    The calculator doesn't have enough digits, so there is no enigma there.
    But why does Excel round? I think the last digit has to be 1, and excel just gives about 15 places and then a stack of zeros.
    Can you set excel to give more acuracy? I guess that this is a second question.

    This is it for me. would someone like to add input please.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Jun 2013
    From
    Lebanon
    Posts
    70
    Thanks
    31

    Re: mod 7

    You need a bigger calculator

    3^{56}=523 347 633 027 360 537 213 511 521
    Thanks from Melody2
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,855
    Thanks
    321
    Awards
    1

    Re: mod 7

    Quote Originally Posted by Melody2 View Post
    find the smallest positive integer b which satisfies 3^56=b(mod7)

    I am very new to modular arithmetic and I stubbled upon this question.
    Wouldn't 'b' have a specific solution.
    Asking for the smallest positive integer b confuses me.


    3^{56} - b = 7k  \;\;\text{      Where k is an integer}

    when I put 3^56 into my calculator or into Excel i get a rounded answer. This also surprised me a little.
    The calculator doesn't have enough digits, so there is no enigma there.
    But why does Excel round? I think the last digit has to be 1, and excel just gives about 15 places and then a stack of zeros.
    Can you set excel to give more acuracy? I guess that this is a second question.

    This is it for me. would someone like to add input please.
    Note that 3^6 \equiv 1~\text{mod 7}

    We have 56 = 9 * 6 + 2, so
    3^{56} \equiv \left ( 3^6 \right ) ^9 \cdot 3^2 \equiv 1 \cdot 2 \equiv 2~\text{mod 7}

    If you want all the digits of 3^(56), see here. It's free.

    -Dan
    Last edited by topsquark; November 24th 2013 at 08:44 AM.
    Thanks from Melody2
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Nov 2013
    From
    Australia
    Posts
    187
    Thanks
    38

    Re: mod 7

    Thanks Idea
    I responded to you in the wee hours but I must not have pressed 'reply'
    I was thinking of writing Santa a note, but now Dan has shown me I don't have to.
    (Unless I am visiting the regional Australia where there is no internet coverage.)
    Melody
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Nov 2013
    From
    Australia
    Posts
    187
    Thanks
    38

    Re: mod 7

    Thanks Dan
    That is exactly what I was hoping for.
    Melody
    Follow Math Help Forum on Facebook and Google+

Search Tags


/mathhelpforum @mathhelpforum