Suppose you have a large collection of rare stamps: each is either a 5-cent stamp or an 8-cent stamp. You urgently need to send a letter, and decide to dip into your collection. What amounts of postage can you make exactly?
Can you say anything about the amounts of postage can you make if the number of each type of stamp you use is even? For example, can you make 97 cents of postage? 103 cents?
Think of the statement having the form, “if you use an even number of 5-cent and 8-cent stamps, then the amount of postage you can make is .” Prove this statement, clearly indicating the style of proof you are using.
Prove that if you have made an even amount of postage, then you must have used an even number of at least one of the two types of stamps. Again, indicate the style of proof you use.
You discover that your letter needs exactly 72 cents of postage, and have found a way to get this exactly. Prove that you must have used at least 6 of one of the types of stamps. Use a different proof technique than you did on the previous parts.
