Number Theory
# Linear Diophantine Equations

In the post office they sell souvenir stamps of three denominations: 25 cents, 10 cents, and 5 cents. If you want to buy 20 stamps for exactly 2 dollars, how many combinations of the three kinds of stamps are there that you can buy?

NOTE: It's possible, in a combination, to buy zero of a stamp variety.

Jane bought 5 watermelons, 3 apples and 20 cherries for 46 dollars. Lizzy bought 2 watermelons, 5 apples and 12 cherries for 27 dollars. If the prices of a watermelon and an apple are both integers, what is the price of an apple?

Give your answer in dollars.