If a and b are relatively prime positive integers, prove that the Diophantine equation ax − by = c has infinitely many solutions in the positive integers.
[Hint: There exist integers x0 and y0 such that ax0 + by0 = c. For any integer t, which is larger than both |x0 | / b and | y0 | /a, a positive solution of the given equation is x = x0 + bt, y = −(y0 − at).]
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