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Commentary on Proposition VII.36¶

Euclid’s rule for finding the L.C.M. of three numbers $a, b, c$ is the rule with which we are familiar. The L.C.M. of $a, b$ is first found, say $d$, and then the L.C.M. of $d$ and $c$ is found. Euclid distinguishes the cases (1) in which $c$ measures $d$, (2) in which $c$ does not measure $d$. We need only reproduce the proof of the general case (2). The method is that of reductio ad absurdum.

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

Since $a, b$ both measure $d$, and $d$ measures $e$,

$a, b$ both measure $e$.

So does $c$.

Therefore $e$ is some common multiple of $a, b, c$. If it is not the least, let $f$ be the L.C.M. Now $a, b$ both measure $f$; therefore $d$, their L.C.M., also measures $f$. [VII. 35]

Thus $d, c$ both measure $f$; therefore $e$, their L.C.M., measures $f$: [VII. 35]

which is impossible, since $f < e$.

Therefore etc.

The process can be continued ad libitum, so that we can find the L.C.M., not only of three, but of as many numbers as we please.