Suppose S is the set of all positive integers. For a,b∈S, define
a∗b=gcd(a,b)lcm[a,b]
For example 8∗12=6.
Show that exactly two of the following three properties are satisfied:
(i) If a,b∈S, then a∗b∈S.
(ii) (a∗b)∗c=a∗(b∗c) for all a,b,c∈S.
(iii) There exists an element i∈S such that a∗i=a for all a∈S. number theoryleast common multiplegreatest common divisor