Two integers are relatively prime (also called coprime) if they have no common positive factors other than 1. In simpler terms, the only positive integer that divides both numbers evenly is 1.
Understanding Relatively Prime Numbers
- Definition: Integers a and b are relatively prime if their greatest common divisor (GCD) is 1, written as gcd(a, b) = 1.
- Key Concept: This does not mean that the numbers are prime themselves. Either or both of the numbers can be composite (non-prime). They only need to share no common factors other than 1.
Examples
- Relatively Prime: 8 and 15 are relatively prime. The factors of 8 are 1, 2, 4, and 8. The factors of 15 are 1, 3, 5, and 15. The only common factor is 1.
- Not Relatively Prime: 12 and 18 are not relatively prime. They share common factors of 1, 2, 3, and 6. Therefore, their GCD is 6, not 1.
How to Determine if Numbers are Relatively Prime
- Find the factors: List all the positive factors of each number.
- Identify common factors: Determine the factors that both numbers share.
- Check the GCD: If the greatest common factor (GCD) is 1, the numbers are relatively prime. You can also use the Euclidean Algorithm to efficiently find the GCD of two numbers.
Why Relatively Prime Numbers are Important
Relatively prime numbers have important applications in:
- Cryptography: Used in various encryption algorithms.
- Number Theory: Fundamental to many theorems and proofs.
- Computer Science: Useful in hashing algorithms and other applications.
In conclusion, relatively prime numbers are integers that share no common positive factors other than 1, a critical concept with significant implications across several fields.
