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Math Help - Classic Diophantine problem

  1. #1
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    Classic Diophantine problem

    For two postage stamp values, one of *n* cents and one of *m* cents
    the following Diophantine equation

    mx + ny = T

    gives the total value of postage or T, where x and y represent the number of each denomination of stamps
    and are therefore integers. Clearly, you could get many values for T.

    But there are also many values of T that you could NOT get.

    I have two questions regarding this situation:

    1) Under what conditions will stamp amounts *m* and *n* generate all but a finite number of
    postage stamp amounts?

    2) Under the conditions of question one (the stamps generate all but a finite number of amounts)
    what is a *formula* for the largest postage value that CANNOT be generated?

    Any help??
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  2. #2
    Super Member PaulRS's Avatar
    Joined
    Oct 2007
    Posts
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    • If (m,n)=M>1 then M divides any linear combination (with natural coefficients) of them, so we do not generate the numbers which are not multiples of M and so there are infinitely many numbers left out.
    • If (m,n)=1, read here
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