Sun Tzu Suan Ching/Examples/Example 2

Example of Problem from

 * There are certain things whose number is unknown.
 * Repeatedly divided by $3$, the remainder is $2$;
 * by $5$ the remainder is $3$,
 * and by $7$ the remainder is $2$.


 * What will be the number?

Solution
The number of objects could be any one of the numbers:
 * $23 + 105 n$

where $n \in \N$ is an arbitrary natural number.

Proof
The numbers in this sequence all leave a remainder of $2$ when divided by $3$:
 * $2, 5, 8, 11, 14, 17, 20, 23, 26, \ldots$

Of these, the numbers in this sequence all leave a remainder of $3$ when divided by $5$:
 * $8, 23, 38, \ldots$

Of these, the numbers in this sequence all leave a remainder of $2$ when divided by $7$:
 * $23, 128, 233, \ldots$

The difference between consecutive terms is $105 = 3 \times 5 \times 7$, according to the Chinese Remainder Theorem.