# Sun Tzu Suan Ching/Examples/Example 2

Jump to navigation
Jump to search
## Example of Problem from

## Example of Problem from *Sun Tzu Suan Ching*

*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.

$\blacksquare$

## Sources

- c. 280 -- 473: Sun Tzu:
*Sun Tzu Suan Ching* - 1992: David Wells:
*Curious and Interesting Puzzles*... (previous) ... (next): Sun Tsu Suan Ching: $70$