On planet Enigma, the residents use a currency called the confusion. There are only 2 confusion bills on Enigma, one worth 8 confusions and the other worth 11 confusions. There are also some coins of smaller value, but each weighs over 10 kilograms, so they are difficult to carry around. In how many ways can a resident of Enigma use only bills to purchase a toaster that costs 96 confusions? (Everybody knows, of course, that vendors on Enigma do not give change, so residents must make their purchases with exact change.) Solution: Since we have two types of bills, we have the Diophantine equation $8a + 11b = 96$, where $a$ and $b$ are non-negative integers. By inspection, we find the solution $a = 12$ and $b = 0$. Note that once we have found a particular solution $(a,b)$, we can decrease $a$ by 11 and increase $b$ by 8 to obtain another solution, because the overall increase in $8a + 11b$ is $8 \cdot (-11) + 11 \cdot 8 = -88 + 88 = 0$. We can also increase $a$ by 11 and decrease $b$ by 8. Furthermore, every solution to the equation $8a + 11b = 96$ can be obtained from the solution $(a,b) = (12,0)$ in this way. So, decreasing $a$ by 11 and increasing $b$ by 8 repeatedly, we get the solutions $(1,8)$, $(-10, 16)$, $(-21, 24)$, and so on. Increasing $a$ by 11 and decreasing $b$ by 8 repeatedly, we get the solutions $(23,-8)$, $(34,-16)$, $(45,-24)$, and so on. However, since $a$ and $b$ are nonnegative, the only solutions are $(12,0)$ and $(1,8)$. There are 2 solutions.

^^first problem (part a) solution