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-   -   Divisibility rules 11 (https://www.askmehelpdesk.com/showthread.php?t=498091)

  • Aug 14, 2010, 10:28 AM
    olenkat
    divisibility rules 11
    I am completing an assignment and had to look for patterns with the number 11.
    I now need to prove 2 of my findings but can't can you help please.
    If I have the number 81752 I know that 8+7+2 -1+5 = 11 so its divisible by 11.
    How do I prove this using albebra
    Also if I take the same number now I know its divisible by 11 and
    8175-2=
    8173
    817-3=
    814
    81-4=
    77 = 7x11 so 7 and the numbers I subtracted being 4,3,2, means 7432x11=81752
    How do I prove this?
  • Aug 14, 2010, 04:25 PM
    galactus

    Wow, cool problem. Mostly it's something like

    "help me solve 1+x=2. How can I find x?".

    Are you familiar with some number theory?

    Also, a number is divisible by 11 only if the alternating sum of its digits sums to 11 or of we obtain 0. That is considered as being divisible by 11 as well. Say, we have 385. We know it is divisible by 11. Alternate the digits and we get 3-8+5=0.

    As for yours:

    8-1+7-5+2=11... check.

    Start with some modular congruencies.



    We can alternate the sign by using odd and even powers of -1:







    Therefore,
    x is divisible by 11 iff the alternating sum of its digits is divisible by 11.
  • Aug 15, 2010, 02:51 AM
    olenkat

    Galactus thanks for helping but my maths isn't so advanced. Can you explain the 3 lines =plus another line. Does that mean 'which means that'. I understand numbers elevated mean to the power of but what do the numbers below a0-a1+a2 mean. And does mod mean multiple of and then I can try translating! Would you be able to help me with the second part of my question. Seeing I only did basic gcse I feel pleased I found what I did but obviously the proving is the difficult part. But thank you any way so far. And also as a first time user I got a buzz that I even got a reply. Are you in the states? Just curious really.
  • Aug 15, 2010, 07:43 AM
    Unknown008

    This symbol: means equivalent.

    As for [math]a_0, a_1, a_2[math] only represent some constant named 'a'. It cold have been any other letter. And the lowered number is there to differentiate between each constant.

    As for the other things, I would be glad to help you, but I don't know about modular congruencies :o
  • Aug 15, 2010, 08:17 AM
    galactus

    I can't give a complete lesson on elementary number theory, but I can explain what the mod business is about.

    We have

    Add the 1 to the 10 and divide by 11.



    Let k=2:





    Say we have



    and so on.

    It's about remainders. It is sometimes called clock arithmetic and is very useful when dealing with huge numbers.

    Arbitrary example:

    Solve

    We want to find all values of x that satisfy the congruence.

    We have gcd(42,90)=6. Since 6 is a factor of 12, there is a solution.

    Solving the congruence is the same as solving

    for integers x and q.

    This reduces to or

    The numbers congruent to 1 modulo 15 are 16,31,46,61,.
    and also -14,-29,-44,.

    See, they're all off by 1 from being a multiple of 15.

    7 is a factor of -14, so we multiply both sides by -2 since

    Thus, we have

    The solution is

    Check: . 90 divides into 450.

    I know this is a lot to digest, but a lot of proofs come from this area of number theory. It is a fascinating field and can be used to solve many problems.
  • Aug 15, 2010, 08:50 AM
    Unknown008

    Uh huh.. I could understand some things, but not everything :(

    I guess I'll have to wait university or next year during my 'spare' time.
  • Aug 15, 2010, 08:57 AM
    galactus
    Quote:

    Originally Posted by Unknown008 View Post
    Uh huh.. I could understand some things, but not everything :(

    I guess I'll have to wait university or next year during my 'spare' time.

    Yes, when you get to school you will no doubt take number theory.
    Then, you will lern about this stuff. You can always get a book or Google. There is lots out there. Just Google "modular arithmetic".
    Or the Chinese Remainder Theorem. That is a goodie.

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