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I have a problem with some basic number theory, would appreciate some help! Watch

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    Hey all,

    Given 4a + 9b = n where a and b are nonnegative integers, I need to prove that for n>23 where n is a natural number the equation is always satisfied

    In other words,

    4a + 9b = 24
    4a + 9b = 25
    4a + 9b = 26
    ....

    will have a set of nonnegative integer solutions (a,b) for n>23.

    Why is it that n has to be greater than 23, not 22 or 24?

    I feel like this has to do with the fact that 4 and 9 are coprime but that is all I could think of.

    I spend hours trying to figure out how to work this out but I can't seem to figure out so I would ask for some help here..
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    (Original post by standkakao)
    Hey all,

    Given 4a + 9b = n where a and b are nonnegative integers, I need to prove that for n>23 where n is a natural number the equation is always satisfied

    In other words,

    4a + 9b = 24
    4a + 9b = 25
    4a + 9b = 26
    ....

    will have a set of nonnegative integer solutions (a,b) for n>23.

    Why is it that n has to be greater than 23, not 22 or 24?

    I feel like this has to do with the fact that 4 and 9 are coprime but that is all I could think of.

    I spend hours trying to figure out how to work this out but I can't seem to figure out so I would ask for some help here..
    May not be the most elegant method.

    Any integer can be written in the form 4k+i, where i\in\{0,1,2,3\}

    Suppose there is a solution pair (a,b), for 4k+i, then is there a solution pair for 4(k+1) +i ?

    Can you take it from there?

    As to n>23: Is there a solution for n=23?
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    https://www.cs.cmu.edu/~adamchik/21-...ty_5_print.pdf

    Page 6,7 might help
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    (Original post by standkakao)
    Hey all,

    Given 4a + 9b = n where a and b are nonnegative integers, I need to prove that for n>23 where n is a natural number the equation is always satisfied

    In other words,

    4a + 9b = 24
    4a + 9b = 25
    4a + 9b = 26
    ....

    will have a set of nonnegative integer solutions (a,b) for n>23.

    Why is it that n has to be greater than 23, not 22 or 24?

    I feel like this has to do with the fact that 4 and 9 are coprime but that is all I could think of.

    I spend hours trying to figure out how to work this out but I can't seem to figure out so I would ask for some help here..
    The reason why n>23 is because of the chicken nugget theorem (it has a proper name, but I have no idea what it is). It just says that the largest integer not expressible as ax+by for fixed positive coprime integers a,b and non negative integers x,y is ab-a-b. In fact, your question is basically just the chicken nugget theorem, so if you want more info, that's what to search for.
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    https://www.artofproblemsolving.com/...Nugget_Theorem


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