Why teachers do not use Least common divisor and Greatest common multiple any more?

Answer 1

I don't know that they ever did. The least common factor of any set of integers is 1. The greatest common multiple of any set of integers is infinite. Values that are the same for every set of numbers are useless for differentiating between them.

Answer 2

It is not that they do not use them any more - they never used them.

The least common divisor of any set of integers is always 1, so it serves no purpose.

And, there is really no such thing as a "greatest common multiple". Once you find the least common multiple of a set of numbers, you can keep adding the LCM to itself over and over again. Each new number you get will be a common multiple of your set of numbers, but each new number will always be larger than the previous. This means that you can keep adding while the number approaches infinity and you will still never find a greatest multiple.