What is co-prime? Give examples of five pairs of co-primes. Are co-prime are always positive? If not, illustrate your answer by example.

Co-prime numbers, also known as relatively prime numbers, are pairs of integers that have no common positive integer factors other than 1. In simpler terms, two numbers are co-prime if their greatest common divisor (GCD) is 1. This concept is fundamental in number theory and has various applications in areas such as cryptography and simplifying fractions.

Identifying Co-prime Pairs

To illustrate what co-prime numbers are, let’s look at some examples. Here are five pairs of co-prime numbers:

• (8, 15) - 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.
• (9, 28) - The factors of 9 are 1, 3, and 9; the factors of 28 are 1, 2, 4, 7, 14, and 28. Again, the only common factor is 1.
• (14, 25) - The factors of 14 are 1, 2, 7, and 14; the factors of 25 are 1, 5, and 25. The only common factor is 1.
• (21, 22) - The factors of 21 are 1, 3, 7, and 21; the factors of 22 are 1, 2, 11, and 22. The only common factor is 1.
• (35, 64) - The factors of 35 are 1, 5, 7, and 35; the factors of 64 are 1, 2, 4, 8, 16, 32, and 64. The only common factor is 1.

Are Co-primes Always Positive?

Co-prime numbers are not restricted to positive integers. In fact, negative integers can also be co-prime. The key point is that the GCD of the two numbers must still be 1. For example, consider the pair (-3, 4):

• The factors of -3 are -1, -3, 1, and 3; the factors of 4 are 1, 2, and 4. The only common factor is 1.

Thus, (-3, 4) is a pair of co-prime numbers. Similarly, the pair (6, -5) is also co-prime because their GCD is still 1, as the factors of 6 are 1, 2, 3, and 6, while the factors of -5 are -1, -5, 1, and 5.

Understanding the Importance of Co-primes

Co-prime numbers play a significant role in various mathematical concepts. For instance, when simplifying fractions, if the numerator and denominator are co-prime, the fraction is already in its simplest form. Additionally, in cryptography, co-prime numbers are used in algorithms like RSA, which relies on the properties of prime numbers and their relationships.

In summary, co-prime numbers are pairs of integers that share no common factors other than 1, and they can be both positive and negative. Understanding co-primes is essential for grasping more complex mathematical concepts and applications.