Let's start! Suppose you have as many postage stamps as you want numbered 2¢ and 3¢. What denominations of postage can you get by combining those? (Or, perhaps, what denominations can you not get with just these two kinds?)

Let's try the same problem with 2¢ and 4¢ stamps. What is the same, what is different?

What about with 3¢ and 4¢ stamps? (What we are really asking, which might be clear by now, is what positive integers \(n\) are impossible to write \(n=3x+4y\) for some nonnegative integers \(x\) and \(y\).) In this case, after some experimentation, it looks like only 1, 2, and 5 are not possible, so anything six or above is possible. We call this number the conductor of the set \(\{3,4\}\). (This is also sometimes called the Frobenius or coin problem.)

Let's spend some time trying this with different small pairs of numbers. Pay attention to two things:

• What is the conductor of the pair? (You might want to ask whether there is such a number!)
• How many numbers lower than the conductor cannot be written in this way?