Least Number with Three Given Fractions

Theorem

 * To find the (natural) number which is the least that will have givenparts.

Proof
Let $a, b, c$ be the given parts.

Let $d, e, f$ be the numbers called by the same name as the parts $a, b, c$.

From, let:
 * $g = \operatorname{lcm} \left\{{d, e, f}\right\}$

So $g$ has parts called by the same name as $d, e, f$.

Therefore $g$ has the parts $a, b, c$.

Suppose there exists $h \in \N: h < g$ which has the parts $a, b, c$.

By, $h$ will be measured by numbers called by the same name as the parts $a, b, c$.

Therefore $h$ is measured by $d, e, f$.

But $h < g$ which is impossible.

Therefore there is no number less than $g$ which has the parts $a, b, c$.