Sun Tzu Suan Ching/Examples/Example 2

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

From Chinese Remainder Theorem: $x \equiv 2 \pmod 3, 3 \pmod 5, 2 \pmod 7$ we have that:

$x \equiv 23 \pmod {105}$

The result follows.

$\blacksquare$


Sources

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