Abstract Algebra - Natural Numbers Proof

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SUMMARY

The discussion centers on the closure of sets of natural numbers under addition, specifically examining sets formed by co-prime numbers such as 3 and 5. It is established that while odd numbers are not closed under addition, sets of multiples are. The professor emphasizes the role of the greatest common divisor (gcd) and the Euclidean Algorithm in determining the closure properties of these sets. As demonstrated, sets formed by co-prime numbers eventually encompass all natural numbers, illustrating a pattern of regularity in their sums.

PREREQUISITES
  • Understanding of natural numbers and their properties
  • Familiarity with the concept of closure in mathematics
  • Knowledge of the Euclidean Algorithm
  • Basic understanding of greatest common divisor (gcd)
NEXT STEPS
  • Study the properties of closure in various mathematical sets
  • Learn how to apply the Euclidean Algorithm to find gcd
  • Explore the implications of co-prime numbers in number theory
  • Investigate the relationship between addition and set regularity in natural numbers
USEFUL FOR

Mathematics students, educators, and anyone interested in number theory and the properties of natural numbers.

adamsmc2
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The question is which sets of natural numbers are closed under addition. I know that odd is not, and I know how to prove that sets of multiples are, but my professor said there is something more and that is has to do with greatest common divisor. He said to pick numbers like 3 and 5 or 5 and 8, then expand the set.

For example 3,5 would be {3,5,6,8,9,10,11,12,13,14,15,...}

He said we should be able to observe over multiple sets something that is not completely obvious but I can't see anything. Possibly it has something to also do with the Euclidian Algorithm but I'm not so sure about that. Also, he said something about when the set starts to show regularity.

Any insight will help. Thanks!
 
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I think you are on the right track. Note that you happened to choose two co-prime numbers. Do you see how at some point, you start to get all numbers (10, 11, 12, ...)?
Given 3 and 5, if you perform the Euclidean Algorithm, that basically gives you ... what?
 
Let n be any natural number

If a is in the set a*n must be
If a and b are in the set, gcd(a,b)*n is, if n is large enough.
 

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