Consider the poset ({3,5,9,15,24,45},/). Which of the following is correct for…

Definitions: greatest element — an element that is divisible by every element of the set (every element divides it).

least element — an element that divides every element of the set.

• Check for a least element: a least element would have to divide 3, 5, 9, 15, 24, and 45.

  No candidate divides both 3 and 5: for example, 3 does not divide 5, and 5 does not divide 3 or 9 or 24. Therefore there is no element that divides every element of the set, so no least element exists.

• Check for a greatest element: a greatest element would have to be divisible by 3, 5, 9, 15, 24, and 45.

  Because 24 is in the set, a greatest element would need to be a multiple of 24, but no element in the set is a multiple of 24. Hence there is no element divisible by every element of the set, so no greatest element exists.

Conclusion: Neither a greatest element nor a least element exists in the poset ({3, 5, 9, 15, 24, 45}, divides).