Coprime numbers, also known as relatively prime numbers, are pairs of numbers that have no common factors other than 1. This means their greatest common divisor (GCD) is 1. Rather than listing all pairs of coprime numbers from 1 to 100, which would be extensive, we can provide examples and clarify the concept.
Understanding Coprime Numbers
Two numbers are coprime if their greatest common factor is 1. This means they share no common prime factors.
Examples of Coprime Pairs
- (2, 3): The factors of 2 are 1 and 2. The factors of 3 are 1 and 3. Their only common factor is 1, making them coprime.
- (7, 12): The factors of 7 are 1 and 7. The factors of 12 are 1, 2, 3, 4, 6, and 12. Their only common factor is 1, so they are coprime.
- (15, 16): The factors of 15 are 1, 3, 5 and 15. The factors of 16 are 1, 2, 4, 8 and 16. Their only common factor is 1, so they are coprime.
Examples of Non-Coprime Pairs
- (4, 6): The factors of 4 are 1, 2, and 4. The factors of 6 are 1, 2, 3, and 6. They share common factors of 1 and 2 so they are not coprime.
- (10, 15): The factors of 10 are 1, 2, 5, and 10. The factors of 15 are 1, 3, 5, and 15. They share common factors of 1 and 5, therefore, they are not coprime.
Selected Coprime Pairs from 1 to 100
The provided reference gives some coprime examples which are listed below as an illustration of coprime pairs:
- (2,3)
- (3,5)
- (5,7)
- (11,13)
- (17,19)
- (21,22)
- (29,31)
- (41,43)
- (59,61)
- (71,73)
- (87,88)
- (99,100)
- (28,57)
- (13,14)
Importance of Coprime Numbers
Coprime numbers play a significant role in several mathematical concepts, such as:
- Modular Arithmetic: They are crucial in understanding modular operations and inverse elements.
- Cryptography: Coprime numbers are fundamental in many encryption algorithms.
- Number Theory: They are important in proofs and theorems regarding prime numbers.
Key Takeaway
It's not practical to list all coprime pairs from 1 to 100. However, understanding the principle that any two numbers whose greatest common factor is 1 are considered coprime, we can identify such pairs.
