mathschallenge.net

Problem

A small village post office is selling 1st class stamps and 2nd class stamps in a book costing £1. If a 1st class stamp is 9p and a 2nd class stamp is 7p, how many ways can a £1 book be filled?

Solution

We are attempting to solve the Diophantine equation 9a + 7b = 100, of which there are two positive solutions:
1 9 + 13 7 = 100
8 9 + 4 7 = 100

How many solutions are there to the equation 4a + 3b = 100?
Try different equations and see if you can spot any patterns.