Sometimes we are only interested in solutions to ax + by = c using nonnegative values for x and y.
(a) Explain why the equation 3x + 5y = 4 has no solutions with x ≥ 0 and y ≥ 0.
(b) Make a list of some of the numbers of the form 3x + 5y with x ≥ 0 and y ≥ 0. Make a conjecture as to which values are not possible. Then prove that your conjecture is correct.
(c) For each of the following values of (a, b), find the largest number that is not of the form ax + by with x ≥ 0 and y ≥ 0.
(i) (a, b) = (3, 7)
(ii) (a, b) = (5, 7)
(iii) (a, b) = (4, 11).
(d) Let gcd(a, b) = 1. Using your results from (c), find a conjectural formula in terms of a and b for the largest number that is not of the form ax + by with x ≥ 0 and y ≥ 0? Check your conjecture for at least two more values of (a, b).
(e) Prove that your conjectural formula in (d) is correct.
(f) Try to generalize this problem to sums of three terms ax + by + cz with x ≥ 0, y ≥ 0, and z ≥ 0. For example, what is the largest number that is not of the form 6x + 10y + 15z with nonnegative x, y, z?
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