Irrational numbers disturbed us, but it's time for Transcendentals now!

The definition of LCM says that we need to find the smallest positive value $k$ , such that $\frac { k }{ e }$ and $\frac { k }{ \pi }$ are integers.

Let's assume that $\frac { k }{ e } =p$ and $\frac { k }{ \pi } =q$ so both $p$ and $q$ are non-zero integers.

Dividing the 1st equation by the 2nd, we get $\frac { k\pi }{ ke } =\frac { p }{ q }$

As $k$ is positive and non-zero, we can cancel it.

$\frac { \pi }{ e } =\frac { p}{ q }$

This means that this LCM will exist only if the above equation is possible.

The above equation just means that $\frac { \pi }{ e }$ is a rational number, the authenticity of which is not known, so the existence of this LCM is unknown .

So as per the question, the required option is Does not exist .

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Please upvote if you liked the solution.

The heading perfectly suits the question, as for the answer, here it is:

It is possible to find the LCM of 2 like irrationals, i.e. irrational numbers of the same kind. We can't find theLCM for two different irrational numbers.

Interestingly, you could look at the concept of LCM in another way: as the least upper bound of a set of numbers under the "divides" partial order relation. Since every number divides 0, if you go by this definition then the LCM of any pair of incommensurable numbers is 0.

But sadly this doesn't seem to be covered by any of the standard definitions of LCM, which are to the effect of "smallest positive number that is an integer multiple of the given numbers" rather than based on LUB by the partial order.