Hermitia is formed as the peaceful economic union of Parodia and Utopia, neither of which however wanted to give up their own currencies, the BING and the BANG. To simplify matters they agreed to use common coins in three denominations. On each of the coins the common value on one side is printed in BANGs, the currency of Parodia, and on the other side it is printed in BINGs, the currency of Utopia. The three coins are the following: the filler worth 3 bangs and 1 bing, the kuna worth 7 bangs and 2 bings, and the taller worth 15 bangs and 4 bings.

Stores are required to post the prices of items as pairs of nonnegative integers. For instance, a pencil may be sold for (11,3), meaning its bang price is 11 and its bing price is 3. A customer may buy it by giving two kunas to the cashier who returns one filler to the customer: 2(7,2) - (3,1) = (11,3). Assuming that the store has an unlimited number of coins in each of the three nominations, can you buy a large eraser, the price of which is 3 bangs in Parodia, and 5 bings in Utopia? Why and how?

In Hilberton, a smaller town in Hermitia, the municipality forbids the storekeepers to return any money (in order to save the cashiers from bringing in and keeping large quantities of coins for opening; in other words, you can buy there something only if you can pay the exact price, in both currencies). To make this law feasible, the municipality also orders all stores to post price combinations x bang/y bing only for integers satisfying the inequalities 3y x and 4x 15y. Is it indeed possible to pay for any item, the price of which satisfies the municipalitys rule, with an appropriate combination of the above coins? Why? What other nomination(s) would make the municipalitys rule feasible?

[Hint: Be careful and do not jump to conclusions too fast! Take some of the names, as suggestions ...