Results 1 to 2 of 2

Math Help - What is the remainder when 11^ 8577314 is divided by 97?

  1. #1
    Newbie
    Joined
    Nov 2011
    Posts
    16

    What is the remainder when 11^ 8577314 is divided by 97?

    What is the remainder when 11^8577314 is divided by 97?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78

    Re: What is the remainder when 11^ 8577314 is divided by 97?

    Quote Originally Posted by sbacon1991 View Post
    What is the remainder when 11^8577314 is divided by 97?
    Since 97 is prime use Fermat's little theorem. It states that

    a^{p-1} \equiv 1 \text{mod}p so this gives that

    11^{96} \equiv 1 \text{mod}97


    So 8577314 \div 96 = 89347 R 2

    This gives that

    8577314=96 \cdot 89347 + 2

    So

    11^{8577314} \equiv 11^{96 \cdot 89347 + 2} \equiv 11^2 \text{mod}97

    It is all down hill from here
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Remainder when large numbers divided
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: June 27th 2011, 07:13 PM
  2. remainder when divided by 7
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: January 14th 2011, 05:43 AM
  3. what is the remainder of the summation when divided by ten
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: July 24th 2010, 03:51 AM
  4. remainder of [9!(16) + 4311]^8603 divided by 11 ?
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: February 5th 2010, 10:38 AM
  5. remainder of sum, divided by 4
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: November 6th 2008, 01:50 PM

Search Tags


/mathhelpforum @mathhelpforum