Sum of two postage stamps: Part One
This is an academic question that is a little tricker than it appears to be.
Joe has a big box of 5-cent and 6-cent postage stamps. He wants to determine the amounts of total postage he can create using ONLY 5-cent and 6-cent stamps and the amounts that he CANNOT create using only 5-cent and 6-cent stamps.
Joe makes the following partial list of postage amounts he can create with only 5-cent and 6-cent stamps:
5 cents (one 5)
6 cents (one 6)
10 cents (two 5's)
11 cents (one 6 & one 5)
12 cents (two 6's)
15 cents (three 5's)
16 cents (two 5's & one 6)
17 cents (one 5 & two 6's)
18 cents (three 6's)
20 cents (four 5's)
However, he makes the following list of postage totals that he CANNOT make using only 5-cent and 6-cent stamps:
For individual stamp values "m" and "n" (which are positive integers) Joe wants to "generalize" the total amounts that he gets (he wants a general statement or rule that describes the relationship between x stamps of value "m", and y stamps of value "n".)
Can you give a "generalization" and explain why it works?
Under what conditions will the numbers of stamps of value m and the number of stamps of value n generate all but a finite number of postage amounts? (I.e., a certain finite number of total postage values cannot be created but all others can be.)
Under these conditions what is a formula for the largest "impossible" postage amount? I.e., a formula for the largest amount that Joe cannot create.
Thanks for any help anybody can give me.