Notes Project for Math 320: Fundamentals of Mathematics Postage Problem, Part 1.
September 19, 2005
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The problem
A few days ago, I had to send a letter by Priority Mail, which costs $3.85. A regular rstclass letter costs 37 cents for the rst ounce and 23 cents for each additional ounce. My stamp drawer is full of 37-cent and 23-cent stamps, but no other denominations. One way to deal with the situation was to use ten 37-cent stamps and one 23-cent stamp for a total of $3.93 in postage. Thats too much. Maybe its better to use only nine 37s, but then I need three 23s for a total of $4.02. Nope, that wastes even more money. What to do? A reasonable person would just go with the $3.93 solution, forget about it, and get on with the events of the day. Not me. I wanted to get the postage to be exactly $3.85 using only 37s and 23s. Can you help? By the way, I also want to send an Express Mail letter, which costs $13.65.
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Mathematize the problem
Although our problem is stated in terms of postage stamps, it is really a problem about the nonnegative integers N0 . In fact, after we burn o the postage-stamp context, the problem can be stated in purely mathematical terms as follows: Let S be the set S = {23x + 37y|x, y N0 }. Problem 1: Is 385 S? Problem 2: Is 1365 S?
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We might also want to establish some terminology to make it easier to discuss the problem. We say that an integer m is realizable with 23, 37, if m S. So now our problems can be stated like this: Problem 1: Is 385 realizable with 23, 37? Problem 2: Is 1365 realizable with 23, 37? One other before thing we get down to business. There is nothing special about the integers 23 and 37. The same type of problems can be posed using any two (or more) positive integers. So if a, b are positive integers, dene S(a, b) = {ax + by|x, y N0 }. And we say that an integer m is realizable with a, b, if m S(a, b). At this point we havent discovered anything about whether an integer is realizable, but at least we can talk about it.
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Getting you feet wet
The number 23 and 37 are too big to start with. Instead, lets deal with some easier cases. Questions: For the True/False questions, supply a proof if True, or a counterexample if False. 1. Which integers are realizable with 1, 2? 2. Which integers are realizable with 1, 500? 3. Which integers are realizable with 2, 3? 4. Which integers are realizable with 4, 6? 5. Which integers are realizable with 2, 101? 6. Which integers are realizable with 3, 5? 7. True or False: If m is rea...