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How does one find the number of integral solutions of equations like,

$2x + 5y + 4z = 20$

I know how to do this counting if the coefficient of every variable on the LHS is $1$. But I don't know how to do this counting with arbitrary coefficients.

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    Do you mean *positive* integral solutions? (Or nonnegative?) There are infinitely many integral solutions.2011-03-25
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    I don't know if this helps, but what you have there is a linear Diophantine equation in which Bézout's identity applies (http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity). It doesn't give any guidance with respect to counting though.2011-03-25

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