First, we need a Diophantine equation. We will work in numbers of cents. Let \(x\) be the number of \(5\)-cent stamps, and \(y\) be the number of 8-cent stamps. We have:

\begin{equation*} 5x + 8y = 637\text{.} \end{equation*}

Convert to a congruence and solve:

\begin{equation*} \begin{aligned}8y \amp \equiv 637 \pmod{5}\\ 3y \amp \equiv 2 \pmod 5\\ 3y \amp \equiv 12 \pmod 5\\ y \amp \equiv 4 \pmod 5. \end{aligned} \end{equation*}

Thus \(y = 4 + 5k\text{.}\) Then \(5x + 8(4+5k) = 637\text{,}\) so \(x = 121 - 8k\text{.}\)

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 \(k\text{.}\)

\(k\) \((x,y)\) Stamps
-1 (129, -1) not possible
0 (121, 4) 125
1 (113, 9) 122
2 (105, 13) 119
\(\vdots\) \(\vdots\) \(\vdots\)

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 have the least number of 5-cent stamps. To minimize the number of 5-cent stamps, we want to pick \(k\) so that \(121-8k\) is as small as possible (but still positive). When \(k = 15\text{,}\) we have \(x = 1\) and \(y = 79\text{.}\)

Therefore, to make $6.37, you can us 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).