Are Relatively Prime Numbers Always Coprime?

In summary, relatively prime numbers are always coprime. This means that they do not share any common factors other than 1. This is because the greatest common divisor of two relatively prime numbers is always 1, making them mutually prime. This property is important in number theory and has applications in cryptography and other mathematical fields. However, it is worth noting that not all coprime numbers are relatively prime, as two numbers can be coprime even if they are not prime themselves. Overall, the concept of relatively prime numbers is a fundamental one in mathematics, with many interesting properties and applications.
  • #1
bhavinsinh
I stumbled across this question:Suppose that a and b are relatively prime.Prove that ab and a+b are relatively prime.
 
Physics news on Phys.org
  • #2
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.
 
  • #3
Or you could assume that d|ab and prove that then d can't divide a+b (d=/=1).
 

1. What are relatively prime numbers?

Relatively prime numbers are two positive integers that have no common factors other than 1. This means that their greatest common divisor (GCD) is equal to 1.

2. How do I know if two numbers are relatively prime?

To determine if two numbers are relatively prime, you can find their GCD using methods such as prime factorization, Euclid's algorithm, or the extended Euclidean algorithm. If the GCD is 1, then the numbers are relatively prime.

3. Can two prime numbers ever be relatively prime?

Yes, two prime numbers can be relatively prime. This is because prime numbers only have 1 and themselves as factors, so they will never have any common factors other than 1.

4. What are some examples of relatively prime numbers?

Examples of relatively prime numbers include 7 and 15, 12 and 25, and 3 and 10. These numbers have no common factors other than 1, so they are relatively prime.

5. Why are relatively prime numbers important in mathematics?

Relatively prime numbers are important in mathematics because they have several applications in number theory, cryptography, and other areas of mathematics. They are also used in the construction of fractions and rational numbers, and in determining if two numbers are co-prime.

Similar threads

  • Linear and Abstract Algebra
Replies
2
Views
761
Replies
5
Views
315
  • Set Theory, Logic, Probability, Statistics
Replies
17
Views
292
  • Programming and Computer Science
Replies
22
Views
636
  • Linear and Abstract Algebra
Replies
11
Views
1K
  • Linear and Abstract Algebra
Replies
17
Views
1K
Replies
8
Views
264
  • Precalculus Mathematics Homework Help
Replies
2
Views
852
  • Linear and Abstract Algebra
Replies
1
Views
800
  • Linear and Abstract Algebra
Replies
4
Views
1K
Back
Top