where a, b, and m are integers. This equation is known as a linear Diophantine equation. Since x and y are required to be integers, the solution is not as simple as you might think.

The following problems involve solutions to linear Diophantine equations.

1. One egg timer can time an interval of exactly 4 minutes, and a second one can time an interval of exactly 7 minutes. Explain how to use the two timers so that an egg can be boiled for exactly 5 minutes.
2. (Bhaskara) The quantity of rubies without flaw, sapphires, and pearls belonging to one person is five, eight and seven respectively; the number of like gems appertaining to another is seven, nine and six; in addition, one has ninety-two gold coins and the other sixty-two and they are equally rich. Tell me quickly then, intelligent friend, who are conversant with algebra, the prices of each sort of gem.
3. (Bachet) A party of 41 persons, men, women, and children, take part in a meal at the inn. The bill is 40 sous and each man pays 4 sous, each woman 3, and each child 1/3 sou. How many men, women, and children were there?