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GCD, Mods and Remainders

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A) Ann visited her grandmother who used to collect stamps. She had a shoebox full of 5 cents, 6 cents, and 7 cents. Ann thought, "I could mail a lot of different size letters and postcards with these stamps." She tried to figure out the different amounts of postage she could make with a combination of those stamps. What totals could she make and what totals are impossible?
b) Ann's dad had shoeboxes with other stamps. One shoebox has 4 cents, 6 cents, and 8 cents stamps. Which totals can Ann make with these stamps? What totals are impossible? Another shoebox has 3 cents, 6 cents, and 8 cents stamps. Which totals can Ann make with these stamps? Which totals are impossible?
c) Generalize for stamp amounts m, n, r where m, n, and r are positive integers. Explain why your generalization works.
d) When there is a largest impossible amount that can be made from a set of stamps, explain how you can determine what the largest impossible amount will be.

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