## Ordering by divisibility

Consider the natural numbers, (partially) ordered by divisibility. Determine the supremum and infimum of the following set: $$M=\{12{,}18,48{,}72\}$$.
An upper bound of the set $$M$$ is any number that can be divided without remainder by every element of $$M$$. The smallest such upper bound is the least common multiple of the numbers in $$M$$.
A lower bound of the set $$M$$ is any number that evenly divides all numbers in $$M$$. The largest such number is the greatest common divisor of the numbers in $$M$$.
$$\sup M=144$$, $$\inf M= 6$$.