Imagine a country that has only two denominations of currency: $7 bills and $11 bills. How would the economy work in this country?

You may answer any or all of the questions below. Quality and quantity are both important.

Part 1: With only $7 and $11 bills available, it is impossible to make a stack of money worth exactly $3. It is also impossible to make a stack worth $15. However, it is possible to make a stack of money worth $25 by using two $7 bills and one $11 bill.

1. Find all amounts that are impossible to make out of $7 and/or $11 bills.

2. What is the greatest impossible amount? Explain why all amounts beyond that total are possible. Prove that all greater amounts are possible without checking them all.

Part 2: There are two different ways to make $120. For example, you can use three $7 bills and nine
$11 bills or you can use fourteen $7 bills and two $11 bills

1.  What is the largest amount of money that can be made in exactly one way? Explain why there is only one way to make a stack worth this amount.
2.  What is the smallest amount that can be made in two different ways? What about three different ways? What about four ways? What about five? Is there a pattern?
3.  Using only $7 and/or $11 bills, there are thirteen different ways to make a stack of money worth exactly $1000. Find all of them. Describe the system you used to find them.

Part 3: Although it is impossible to make a pile of money worth exactly $3, it is still possible to buy something that costs $3. You can give two $7 bills to the cashier and receive one $11 bill as change.

1.  Find all amounts that cannot be used as prices because they are impossible to make out of $7 and $11 bills if you can get change when making a purchase.
2. Is there a greatest impossible price beyond which all prices are possible? If not, why not? If so,can you explain why all prices beyond that price are possible?