Sun Tzu Suan Ching/Examples/Example 1

Example of Problem from

 * A woman was washing dishes in a river, when an official whose business was overseeing the waters demanded of her:
 * "Why are there so many dishes here?"


 * "Because a feasting was entertained in the house," the woman replied.
 * Thereuon the official enquired the number of guests.


 * "I don't know," the woman said, "how many guests there had been;
 * but every $2$ used a dish for rice between them;
 * every $3$ a dish for broth;
 * every $4$ a dish for meat;
 * and there were $65$ dishes in all."

Solution

 * $60$ guests.

Proof
Let $x$ be the number of guests.

Then: