Commentary on Proposition VII.39

Back to Proposition VII.39.

[Folio 0350 | p. 344]

This again is practically a restatement in another form of the problem of finding the L.C.M.

To find a number which has $\frac{1}{a}\text{th}$, $\frac{1}{b}\text{th}$ and $\frac{1}{c}\text{th}$ parts.

Let $d$ be the L.C.M. of $a, b, c$.

Thus $d$ has $\frac{1}{a}\text{th}$, $\frac{1}{b}\text{th}$ and $\frac{1}{c}\text{th}$ parts. [VII. 37]

If it is not the least number which has, let the least such number be $e$. Then, since $e$ has those parts, $e$ is measured by $a, b, c$; and $e < d$: which is impossible.