Results 1 to 1 of 1

Math Help - Prove divisibility test for 11 using induction

  1. #1
    MHF Contributor alexmahone's Avatar
    Joined
    Oct 2008
    Posts
    1,074
    Thanks
    7

    Prove divisibility test for 11 using induction

    Consider the positive integer 11.
    1 - 1 = 0, which is divisible by 11.

    Let the statement be true for a positive integer m, which is divisible by 11.
    S(m)=a_1-a_2+a_3-...+(-1)^{n-1}a_n is divisible by 11.

    Consider the positive integer m + 11.

    If the last digit of m is <= 8, it will increase by 1. If i digits before the last digit are equal to 9, they will all become 0 in m + 11. The digit before the i digits (equal to 9) will increase by 1, with the result that S (m + 11) = S (m) if i is even and S(m+11)=S(m)\pm9\pm2=S(m)\pm11 if i is odd.

    If the last digit of m is 9, the last digit of m + 11 becomes 0. If the second last digit of m is < 8, it increases by 2, with the result that S(m+11)=S(m)\pm9\pm2=S(m)\pm11. If, however, the second last digit of m is 8 or 9 it becomes 0 or 1 respectively. If i digits before the second last digit are equal to 9, they all become 0 and the digit before these i digits increases by 1, with the result that S(m+11)=  S(m)\pm9\mp8\mp1=S(m) if i is even and S(m+11)=S(m)\pm9\mp8\pm9\pm1=S(m)\pm11 if i is odd.

    So, the sum of digits of n + 11 is also divisible by 11.
    This completes the proof.

    Could someone please check my proof and see if I have made any mistakes or missed out any cases? Thanks.
    Last edited by alexmahone; August 13th 2011 at 10:17 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Prove divisibility test for 3 using induction
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: August 12th 2011, 11:03 AM
  2. Induction divisibility P1
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: February 6th 2010, 10:20 AM
  3. Prove divisibility by induction
    Posted in the Algebra Forum
    Replies: 9
    Last Post: June 2nd 2009, 12:14 PM
  4. Prove divisibility by 7 (using induction)
    Posted in the Algebra Forum
    Replies: 1
    Last Post: October 29th 2008, 11:12 AM
  5. Prove divisibility by induction
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: July 30th 2008, 12:58 PM

Search Tags


/mathhelpforum @mathhelpforum