For such a monoid, if J [subset] S is such that J has a common multiple, then a least common multiple (lcm) exists and is unique.

We will call cliques the subsets of S having a common multiple, and let J be the set of all cliques; if J is a clique, we note [M.

It is indeed a special case of our Theorem 2: in trace monoids, for a subset J of S, only two disjoint cases can occur: either all elements of J commute, and their product is clearly their least common multiple; or there exist two elements of J which do not commute, and J does not admit a common multiple.

4 of [17]) and has the least common multiple property (see Brieskorn and Saito, Proposition 4.

Let L(n) denotes the least

common multiple of all positive integers from 1 to n, then we have

1] Amarnth Murthy, Some notions on least

common multiples, Smarandache Notions Journal, 12(2001), 307-309.

In fact, from the properties of the least

common multiple of any positive integers, we know that

Others are bluntly saying that their revolutionary rights are indisputable, and therefore the fellow liberators can use anything including corruption as the least

common multiples just to quench their thirst of liberation struggle on the expense of public funds The freedom fighters in South Sudan right from the Anyanya revolutionary movement and the current Sudan People Liberation Movement and Army SPLM/A could not develop our national identity, to which was the cause of civil conflict.

1] Amarnath Murthy, Some Notions on Least

Common Multiples, Smarandache Notions Journal, 12(2004), 307-308.

Higher level strategies are developed by using objects to obscure part of the picture and by examining common multiples and factors.

Equivalence: To develop a foundation for later algebra, students use the Snap Blocks virtual manipulative to explore equivalent expressions, common factors, and common multiples.