Give examples of coprime where the numbers are (i) primes (ii) composite (iii) prime and composite.

Coprime numbers, also known as relatively prime numbers, are pairs of numbers that have no common positive integer factors other than 1. This means their greatest common divisor (GCD) is 1. Let's delve into examples of coprime numbers across different categories: primes, composites, and a mix of both.

Examples of Coprime Numbers

Coprime Primes

When we consider prime numbers, any two distinct primes are coprime. For instance:

• 3 and 5: Both are prime numbers, and their only common factor is 1.
• 7 and 11: Again, both are primes with no common factors other than 1.

This property holds true because prime numbers are defined as having exactly two distinct positive divisors: 1 and the number itself. Therefore, distinct primes will always be coprime.

Coprime Composites

Moving on to composite numbers, which have more than two factors, we can still find pairs that are coprime. Consider the following examples:

• 8 and 9: The factors of 8 are 1, 2, 4, and 8, while the factors of 9 are 1, 3, and 9. Their only common factor is 1.
• 14 and 25: The factors of 14 are 1, 2, 7, and 14, and the factors of 25 are 1, 5, and 25. Again, the only common factor is 1.

In these cases, even though both numbers are composite, they do not share any prime factors, making them coprime.

Coprime Prime and Composite

Lastly, let's look at pairs that consist of one prime and one composite number:

• 5 and 12: The prime number 5 has factors of 1 and 5, while the composite number 12 has factors of 1, 2, 3, 4, 6, and 12. Their only common factor is 1.
• 3 and 14: Here, 3 is prime and has factors of 1 and 3, while 14 has factors of 1, 2, 7, and 14. They share no common factors other than 1.

This illustrates that a prime number can be coprime with a composite number as long as they do not share any prime factors.

Understanding the Importance of Coprime Numbers

Coprime numbers play a significant role in various areas of mathematics, including number theory, cryptography, and even in simplifying fractions. Recognizing coprime pairs can help in solving problems related to ratios, proportions, and modular arithmetic.

In summary, whether dealing with primes, composites, or a combination of both, the key takeaway is that coprime numbers are defined by their lack of common factors beyond 1. This property is fundamental in many mathematical concepts and applications.