Let and be integers. Consider the ring

.

This is a well-ordered group. So by a result in this post it is infinite cyclic. We call the positive generator the *greatest common divisor* (GCD) of and .

Consider now the ring

is divisible by both and .

It is also a well-ordered group. Hence it is generated by a single positive element , called the *least common multiple* (LCM) of and .

Let

and .

, and are subings of . Moreover, and . So by the Chinese remainder theorem,

,

which can be written as

by the third isomorphism theorem. The groups in brackets are all finite groups of orders , , and . Hence , i.e.,

.

In general, let . As before, we can define

is divisible by

for .

Then and . As before,

,

i.e.,

.

So , i.e.

.

If we replace by for any number field , then takes the form

.

Taking cardinalities gives the following equation in terms of ideal norms

.

Thus

.