Co-prime numbers are two or more numbers that share only 1 as their common factor; in other words, their highest common factor (HCF) is 1.
Understanding Co-Prime Numbers
Co-prime numbers (also known as relatively prime or mutually prime numbers) are a set of integers that have no common factors other than 1. This means that when you find the factors of each number in the set, the only number they both share is 1. It's crucial to understand that the numbers themselves don't necessarily have to be prime.
Key Characteristics:
- Common Factor: The only common factor between co-prime numbers is 1.
- HCF/GCD: Their highest common factor (HCF), also known as the greatest common divisor (GCD), is 1.
- Number of Integers: Co-primality involves at least two integers.
Examples of Co-Prime Numbers:
- 4 and 9: The factors of 4 are 1, 2, and 4. The factors of 9 are 1, 3, and 9. Their only common factor is 1.
- 15 and 28: The factors of 15 are 1, 3, 5, and 15. The factors of 28 are 1, 2, 4, 7, 14, and 28. Their only common factor is 1.
- 8, 9, and 25: The factors of 8 are 1, 2, 4, and 8. The factors of 9 are 1, 3, and 9. The factors of 25 are 1, 5, and 25. Their only common factor is 1.
Examples of Non-Co-Prime Numbers:
- 6 and 9: The factors of 6 are 1, 2, 3, and 6. The factors of 9 are 1, 3, and 9. They share the common factor 3 (in addition to 1), so they are not co-prime.
- 12 and 18: Both 12 and 18 share common factors other than 1 (e.g., 2, 3, and 6). Therefore, they are not co-prime.
Distinguishing Co-Prime from Prime Numbers
A prime number is a number greater than 1 that has only two factors: 1 and itself (e.g., 2, 3, 5, 7). Co-prime numbers, on the other hand, are a set of numbers that share only the factor 1. A single number cannot be co-prime; it requires at least one other number to establish a relationship. Furthermore, the individual numbers in a co-prime set do not need to be prime numbers themselves.
