First, we need a Diophantine equation. We will work in numbers of cents. Let be the number of -cent stamps, and be the number of 8-cent stamps. We have:
Convert to a congruence and solve:
Thus Then so
This says that one way to make $6.37 is to take 121 of the 5-cent stamps and 4 of the 8-cent stamps. To find the smallest and largest number of stamps, try different values of
|
|
Stamps |
|
|
|
-1 |
(129, -1) |
not possible |
0 |
(121, 4) |
125 |
1 |
(113, 9) |
122 |
2 |
(105, 13) |
119 |
|
|
|
This is no surprise. Having the most stamps means we have as many 5-cent stamps as possible, and to get the smallest number of stamps would require having the least number of 5-cent stamps. To minimize the number of 5-cent stamps, we want to pick so that is as small as possible (but still positive). When we have and
Therefore, to make $6.37, you can use as few as 80 stamps (1 5-cent stamp and 79 8-cent stamps) or as many as 125 stamps (121 5-cent stamps and 4 8-cent stamps).