Proving lcm(m,n)=k using mZ and nZ intersection

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SUMMARY

The discussion focuses on proving the intersection of mZ and nZ equals kZ, where k is the least common multiple (lcm) of m and n. The user successfully demonstrates that kZ is a subset of both mZ and nZ but struggles to establish the reverse inclusion. The relationship between x, a common multiple of m and n, and k is crucial for completing the proof.

PREREQUISITES
  • Understanding of least common multiples (lcm) and their properties
  • Familiarity with integer multiples and set notation (e.g., mZ, nZ)
  • Basic knowledge of number theory concepts
  • Proficiency in mathematical proof techniques
NEXT STEPS
  • Study the properties of least common multiples in number theory
  • Learn about set intersections and their implications in mathematics
  • Explore proofs involving integer multiples and divisibility
  • Investigate the relationship between common multiples and lcm
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Mathematics students, educators, and anyone interested in number theory and mathematical proofs related to integer sets.

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Homework Statement


prove:[itex]mZ\cap nZ=kZ[/itex]
where lcm(m,n)=k

Homework Equations


The Attempt at a Solution


i can prove that kZ is a subset of mZ and nZ but i cannot prove further!
thx for ur help!
 
Physics news on Phys.org
If x in mZ and nZ, x = am = bn for some integers a and b. So x is a common multiple of m and n. What is the relationship of x to k?
 

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