Are Relatively Prime Numbers Always Coprime?

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SUMMARY

The discussion centers on the mathematical proof that if two integers a and b are relatively prime, then the numbers ab and a+b are also relatively prime. The proof utilizes the relationship (a+b)/(ab) = 1/b + 1/a, demonstrating that any common divisor d of ab must not divide a+b, provided d is not equal to 1. This establishes that the greatest common divisor (gcd) of ab and a+b is 1, confirming their coprimality.

PREREQUISITES
  • Understanding of coprime and relatively prime concepts in number theory
  • Familiarity with basic algebraic manipulation
  • Knowledge of greatest common divisor (gcd) calculations
  • Basic comprehension of mathematical proofs and logic
NEXT STEPS
  • Study the properties of coprime numbers in number theory
  • Explore proofs involving gcd and their applications
  • Learn about the Euclidean algorithm for calculating gcd
  • Investigate the implications of coprimality in modular arithmetic
USEFUL FOR

Mathematicians, students of number theory, educators teaching mathematical proofs, and anyone interested in the properties of integers and their relationships.

bhavinsinh
I stumbled across this question:Suppose that a and b are relatively prime.Prove that ab and a+b are relatively prime.
 
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Use the fact that:
(a+b)/(ab)=1/b+1/a.
If I say more than this, then you wouldn't have anything to do for yourself.
 
Or you could assume that d|ab and prove that then d can't divide a+b (d=/=1).
 

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