## Ordering by divisibility

### Task number: 2827

Consider the natural numbers, (partially) ordered by divisibility. Determine the supremum and infimum of the following set: \( M=\{12{,}18,48{,}72\} \).

#### Resolution

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 \(M\).

#### Result

\(\sup M=144\), \(\inf M= 6\).