《孫子算經卷下》 "Sun Tzŭ's Computational Classic: Volume III"
§35. Lowest common multiple

This section gives a word problem asking for the lowest common multiple.

Translation

Chinese source text: Version A, Version B, Version C, Version D.
Unless noted otherwise, I follow the text from Version D, 《知不足齋叢書》本.

Source text Target text Notes
今有三女、長女五日一歸、中女四日一歸、少女三日一歸。問三女幾何日相會。 Suppose there be three daughters: [the] elder daughter [every] five days returneth once, [the] middle daughter [every] four days returneth once, [and the] younger daughter [every] three days returneth once. [We] ask, [every] how many days meet [the] three daughters with each other?
答曰、六十日。 Answer saith: [every] sixty days.
術曰、置長女五日中女四日少女三日於右方、各列一算於左方。 Method saith: put [the] elder daughter's five days, [the] middle daughter's four days, [and the] younger daughter's three days upon [the] right, [and for] each, rank one rod upon [the] left.
  • In modern notation, form the matrix
    \roundbr{ \begin{array}{cc} 1 & 5 \\ 1 & 4 \\ 1 & 3 \end{array} }.
維乘之、各得所到數、長女十二到、中女十五到、少女二十到。 [In] linkage multiplying them, each resulteth in [the] number of that arrived: [the] elder daughter twelve arrivals, [the] middle daughter fifteen arrivals, [and the] younger daughter twenty arrivals.
  • 維乘之： [in] linkage multiplying them

    This appears to mean taking each of the ones, and multipying it with the non-ones in the other two rows:

    \roundbr{ \begin{array}{cc} 1 \times 4 \times 3 & 5 \\ 1 \times 5 \times 3 & 4 \\ 1 \times 5 \times 4 & 3 \end{array} } = \roundbr{ \begin{array}{cc} 12 & 5 \\ 15 & 4 \\ 20 & 3 \end{array} }.
又各以歸日乘到數、即得。 And each multiplying [the] number of arrivals by [the] days of return, [we] are done.
  • In modern notation,
    \roundbr{ \begin{array}{c} 12 \times 5 \\ 15 \times 4 \\ 20 \times 3 \end{array} } = \roundbr{ \begin{array}{c} 60 \\ 60 \\ 60 \end{array} }.
    This algorithm does not in general give the lowest common multiple. It merely gives the product of the three numbers, which happens to coincide with the lowest common multiple when the three numbers are pairwise coprime.

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume III §35". <https://yawnoc.github.io/sun-tzu/iii/35> Accessed yyyy-mm-dd.