Understanding the Concept of Co-Prime Numbers or Relative Primes

A co-prime number is a pair of numbers with no divisor other than the common one. A set of relatively prime numbers must consist of at least two numbers. The relatively prime number is only one as the greatest common divisor. For example, prime numbers are 4 and 7, 5, 7, and 9. Coprime numbers do not necessarily have to be prime numbers. Coprime is formed from two composite numbers, such as 4 and 9. Co-prime numbers are always two prime numbers. They only have one thing in common. Consider the numbers 29 and 31. 29 has only two prime factors, 1 and 29. 31 has only two prime factors, 1 and 31. 

What is a Co-prime Number?

If the two numbers a and b have only one divisor in common, then a and b are relatively prime. In this case, (a, b) are called disjoint pairs. A Co-prime number is called a disjoint or disjoint number. Read more of this article to learn what is the co-prime number.

What is Co Prime Number: Definition

Coprime is not the same as a prime number. If two numbers have a common divisor, they are called relatively prime numbers. As an example, consider the following: 

 Take two numbers, like 16 and 17.

 Factoring each into the following: 

 16 = 1x2x2x2 = 1x4x4 = 1x2x8 = 1x2x8 = 1x2x8 = 1x2x8 = 1x2x8 = 1x2x 8 = 1  x 17 = 17 

 In this case, the least common multiple (HCM) of the two numbers is 1. Therefore, 16 and 17 are relatively prime to each other. 

 Let’s look at another example,  18 and 24, to see what you can get by factoring both numbers. 

 1x2x3x3 = 18 

 24 = 1 x 2 x 2×3 or 1x4x6 or 1×3 x 8 

 Both numbers have three common elements: “1”, “2”, and “3”. From this, we conclude that 18 and 24 are not disjoint.

Properties of Co-prime Numbers

A Co-prime number can be easily identified by using the properties described below:

• The HCF of two coprime numbers is always 1. Because 5 and 9 are coprime numbers, HCF (5, 9) = 1.

• The product of two coprime numbers is every time their Least Common Multiple (LCM). 5 and 9 are co-prime numbers, for example. As a result, LCM (5, 9) = 45.

• With each number, 1 forms a co-prime number pair.

• Because they always have two as a common factor, two even numbers cannot be co-prime.

• The product of two co-prime numbers is always co-prime with the sum of their co-prime numbers. 5 and 9 are co-prime numbers, for example. In this case, 5 + 9 = 14 is co-prime with 5* 9 = 45.

• Co-prime numbers are all pairs of Two numbers one after another. Two numbers, one after another, have one as their common factor.

Co-prime and Twin Prime Numbers

A co-prime number is a pair of numbers whose HCF of 1. On the other hand, Twin prime numbers are prime numbers whose difference is always 2. Three and five, for example, are twin prime numbers. The following points distinguish co-prime and twin prime numbers.

• Twin prime numbers are always prime numbers, whereas co-prime numbers can also be composite.

• The difference between two twin primes is always two, whereas the difference between two coprimes can be any number.

• All twin prime number pairs are co-prime numbers, whereas all co-prime numbers may or may not be twin primes.

• Every number forms a co-prime pair with 1, but only 3 forms a twin prime pair.

Co-prime Numbers from 1 to 100

Many pairs can be listed as co-prime numbers in the co-prime numbers from 1 to 100 based on the above properties. Co-prime number pairs ranging from 1 to 100 include (1, 2), (3, 67), (2, 7), (99, 100), (34, 79), (54, 67), (10, 11), and so on. Experiment with forming more such pairs of co-prime numbers on your own.

Important Notes

• If the GCF of two numbers is 1, they are co-prime, and vice versa.

• Co-prime numbers don’t need to be prime numbers. 12 and 35, for example, are co-prime numbers, even though 12 and 35 are NOT prime numbers.

Tips and Tricks

• Co-prime numbers are two prime numbers.

• Co-prime numbers are two numbers one after another.

• Any other number and 1 form a co-prime pair.

• Any number that is not its multiple is co-prime with a prime number.

• Two even numbers should never be co-primed.

Conclusion

Two numbers are co-prime to one another if they share no common factors, except for 1. For example, the co-primes of 11 are 5 and 7. Conversely, two numbers are relatively prime to one another if their greatest common factor is 1 (or they are co-prime). For example, the relative primes of 11 and 15 are 5 and 3, respectively. Now you know what a co-prime number is. With this, you will have a clear understanding of co-prime numbers, twin prime numbers and the most important facts related to both of them.