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roto25
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Homework Statement
Prove gcd(lcm(a, b), c) = lcm(gcd(a, c), gcd(b, c))
I've tried coming up with a way to even rewrite it but I'm not really able to do it.
Number theory proof - gcf and lcm
- Thread starter roto25
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- Number theory Proof Theory
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SUMMARY
The discussion focuses on proving the mathematical identity gcd(lcm(a, b), c) = lcm(gcd(a, c), gcd(b, c). Participants suggest using Venn diagrams to visualize the relationships between prime factors, where the least common multiple (lcm) represents a union and the greatest common divisor (gcd) represents an intersection. This approach simplifies the proof process by providing a clear graphical representation of the concepts involved.
PREREQUISITES- Understanding of greatest common divisor (gcd) and least common multiple (lcm)
- Familiarity with prime factorization
- Basic knowledge of set theory concepts, particularly unions and intersections
- Ability to interpret Venn diagrams
- Study the properties of gcd and lcm in number theory
- Learn how to construct and interpret Venn diagrams for mathematical proofs
- Explore prime factorization techniques for integers
- Investigate additional proofs involving gcd and lcm relationships
Students of mathematics, particularly those studying number theory, educators teaching mathematical proofs, and anyone interested in enhancing their understanding of gcd and lcm concepts.
Prove gcd(lcm(a, b), c) = lcm(gcd(a, c), gcd(b, c))
I've tried coming up with a way to even rewrite it but I'm not really able to do it.
- #2
I like Serena
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roto25 said:
Hi roto!
Easiest is to set up a couple of Venn diagrams with the supposed prime factors in it.
An lcm is a union and a gcd is an intersection.
Do you know how to do that?
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