Find the LCM -1 , -1 , 2-3i

Problem

LCM(−1,−1,2−3*i)

Solution

1. Identify the numbers in the set. The numbers provided are −1 −1 and the complex number 2−3*i

2. Understand the definition of Least Common Multiple (LCM) for integers and complex numbers (Gaussian integers). In the context of ring theory, the LCM of a set of elements is an element L such that every element in the set divides L and L divides any other common multiple.

3. Determine the units in the ring of Gaussian integers ℤ*[i] The units are 1, -1, i, -i$. Multiplying by a unit does not change the divisibility properties; therefore, the LCM is only unique up to multiplication by a unit.

4. Analyze the divisibility of the real integer −1 Since −1 is a unit in the ring of Gaussian integers, it divides every element in the ring.

5. Apply the property that if u is a unit, LCM(u,α)=α (up to a unit factor). Since −1 is a unit, the LCM of −1 and any Gaussian integer α is simply α

6. Conclude that the LCM of −1,−1, and 2−3*i is 2−3*i

Final Answer

LCM(−1,−1,2−3*i)=2−3*i

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