A number L is called a common multiple of m and n if both m and n divide L. The smallest such L is called the least common multiple of m and n and is denoted by LCM(m,n). For example, LCM(3,7)=21 and LCM(12,66)=132.

a.) find the following least common multiples.

i) LCM(8,12) ii) LCM(20,30) iii) LCM(51,68) iv) LCM(23,18)

b.) For each of the LCMs that you computed in (a), compare the value of LCM(m,n) to the values of m, n, and gcd(m,n). Try to find a relationship.

c.) give an argument proving that the relationship you found is correct for all m and n.

d.) Use your result in (b) to compute LCM(301337,307829).

e.) Suppose that gcd(m,n)=18 and LCM(m,n)=720. Find m and n. Is there more than one possibility? If so, find all of them.