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Math Help - Sum of unknowns

  1. #1
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    Sum of unknowns

    I've edited the original question because there was a mistake in it.


    can x and y be determined if it is given that there are 65 values which the sum a multiple of x and a multiple of y cannot give

    I.e. Nx + My has 65 unattainable values, where N and M are variables. Also x and y both do not equal 2.

    I saw it on: Hard math questions, again? - Yahoo! Answers

    Its the last part to the last question.
    Last edited by Aquafina; May 14th 2009 at 01:17 AM. Reason: typed the question wrong
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  2. #2
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    Quote Originally Posted by Aquafina View Post
    can x and y be determined if it is given that a multiple of x and a multiple of y added together give 65.

    I.e. Nx + My = 65, where N and M are integers. Also x and y both do not equal 2.
    No, you need another equation
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  3. #3
    Senior Member I-Think's Avatar
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    Diophantine equation

    Salutations.

    A question to e^i*pi, if the values of M and N are given, couldn't you solve this as a Diophantine equation?

    Question (Not relevant) #2: Why does spell-check insist I capitalize 'Diophantine'?
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    Quote Originally Posted by I-Think View Post
    Salutations.

    A question to e^i*pi, if the values of M and N are given, couldn't you solve this as a Diophantine equation?
    Yes, you could based on a simple divisibility condition on the gcd(M,N).

    Question (Not relevant) #2: Why does spell-check insist I capitalize 'Diophantine'?
    Because its named after Diophantus,I guess....
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    Thank you, no M and N are not given. Any reason why it doesn't work without it?
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  6. #6
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    Quote Originally Posted by Aquafina View Post
    Thank you, no M and N are not given. Any reason why it doesn't work without it?
    For such a general case read the "linear diophantine equation" section here.
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    hi thanks, so the reasoning is we can't solve it because we don't know if c is the greatest common divisor of a and b?

    But i thought that is just the case if we want to have infinite solutions?

    i.e. with pentagonal square numbers would have infinite integer solutions, right?

    Pentagonal Square Number -- from Wolfram MathWorld
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    Quote Originally Posted by Aquafina View Post
    hi thanks, so the reasoning is we can't solve it because we don't know if c is the greatest common divisor of a and b?

    But i thought that is just the case if we want to have infinite solutions?
    It suffices if gcd(M,N) divides 65 to have infinite solutions, else it wont have any solution...

    To see this observe that if gcd(M,N) does not divide 65 and there exists a solution, then gcd(M,N) divides the left hand side, but not the right hand side...

    If it does divide, then use division algorithm to solve for Mx+Ny = \text{gcd}(M,N). Let (x_0,y_0) be one solution and then generate x = x_0 + Nt, y = y_0 - Mt for various t, to get infinite solutions.
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  9. #9
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    ah thanks, so that would imply all diophantine equations either have infinite integer solutions or no solutions?

    or is this just for linear ones? i.e. the example i used, x^2 - 6y^2 = 1 is not a correct example
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    Lord of certain Rings
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    Quote Originally Posted by Aquafina View Post
    ah thanks, so that would imply all diophantine equations either have infinite integer solutions or no solutions?

    or is this just for linear ones? i.e. the example i used, x^2 - 6y^2 = 1 is not a correct example
    Its definitely true for all linear ones. You cant say it in general

    Note: The one you quoted is of the form called "Pell's equation". Read the wiki on it if you are interested...
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    Unhappy

    Ah right, I think I'll make a new thread for the Pell's Equations then lol

    Anyways, did u just edit your previous post because this wasnt there before (i think)

    To see this observe that if gcd(M,N) does not divide 65 and there exists a solution, then gcd(M,N) divides the left hand side, but not the right hand side...

    If it does divide, then use division algorithm to solve for . Let be one solution and then generate for various t, to get infinite solutions.


    I havent done algorithms (sorry ) could you explain that paragraph if you dont mind?
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    Error in original question

    oops i just realised im doing the question wrong...

    It is actually:

    can x and y be determined if it is given that there are 65 values which the sum a multiple of x and a multiple of y cannot give

    I.e. Nx + My has 65 unattainable values, where N and M are variables. Also x and y both do not equal 2.


    I saw it on: Hard math questions, again? - Yahoo! Answers

    Its the last part to the last question.
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  13. #13
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    Quote Originally Posted by Aquafina View Post
    Ah right, I think I'll make a new thread for the Pell's Equations then lol

    Anyways, did u just edit your previous post because this wasnt there before (i think)
    Yes

    I havent done algorithms (sorry ) could you explain that paragraph if you dont mind?
    Dont worry about the name.. Its the method you use to compute the gcd..
    Do you know Bezout's identity?
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  14. #14
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    read the post above, question slightly modified
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  15. #15
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    chicken mcnugget theorem..

    I think this requires the chicken mcnugget theorem

    but i'm not sure how to use it. any ideas?

    Chicken McNugget Theorem - AoPSWiki
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