Let m and n be relatively prime integers each greater than
1. Assume you have an unlimited supply of m- and n-cent stamps. Using only these stamps, show that
(a) it is not possible to purchase a selection of stamps worth precisely mn − m − n cents;
(b) for any r > mn − m − n, it is possible to purchase a selection of stamps worth exactly r cents. [Hint: There exist integers a and b, with 0 < a < n and 0 < b < m, such that bn = am − 1 and hence such that (n − a)m = (m − b)n − 1. See Exercise 36 of Section 4.2. Now try downward induction!]
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