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Define a*b=LCM[a,b]/GCD(a,b)

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April 28, 2012
number theoryleast common multiplegreatest common divisor

Problem Statement

Suppose SS is the set of all positive integers. For a,b∈Sa,b \in S, define a∗b=lcm[a,b]gcd(a,b)a * b=\frac{\text{lcm}[a,b]}{\text{gcd}(a,b)} For example 8∗12=68*12=6. Show that exactly two of the following three properties are satisfied: (i) If a,b∈Sa,b \in S, then a∗b∈Sa*b \in S. (ii) (a∗b)∗c=a∗(b∗c)(a*b)*c=a*(b*c) for all a,b,c∈Sa,b,c \in S. (iii) There exists an element i∈Si \in S such that a∗i=aa *i =a for all a∈Sa \in S.
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