Last updated on July 22nd, 2025
Coprime numbers are a set of at least two numbers that share only 1 as a common factor. The highest common factor of a pair of relatively prime numbers is 1, and they are considered coprime. Here, In this article, we will explore the concept of coprime numbers.
Imagine a and b are two numbers. If 1 is the only common factor of both these numbers, then a and b are coprime numbers. Remember that coprime numbers are not always prime numbers. Examples of coprime numbers include (2, 15), (3, 8), and (4, 9). Other than 1, these numbers have no common factors.
Key properties of coprime numbers are:
Coprime numbers can be either composite or prime numbers. However, they differ from each other, and the main differences between prime and coprime numbers are listed below:
Characteristics |
Prime numbers |
Coprime numbers |
Definition |
1 and the number itself are the only two factors of a prime number. These natural numbers are greater than 1. |
Coprime numbers share only 1 as a common factor |
Divisibility |
A prime number is only evenly divisible by 1 and the number itself. |
Coprime numbers share no common factor other than 1. They can be divided evenly only by 1. |
Examples |
2, 3, 5, 7, 11, 13, 17, etc. |
(1, 2), (4, 5), (8,15), (20,21) |
Total numbers |
A prime number is a single number. |
Coprime numbers are a set or pair of two or more numbers. |
Nature of numbers |
Prime numbers are natural numbers greater than 1. |
Coprime numbers are usually defined among positive integers. |
Common factor |
The two factors of a prime number are 1 and the number itself. |
1 is the only common factor of coprime numbers. |
Special feature |
Except for 2, all other prime numbers are odd. |
Coprime numbers are not always prime numbers. For instance, 4 and 5 are coprime numbers( but 4 is not a prime number). |
If there is no other positive integer other than 1 to divide two integers, then the two numbers are coprime. Furthermore, if the greatest common factor of two numbers is 1, then we can consider the numbers as coprime. Take a look at this example. Let us consider two numbers, 6 and 11.
Now we can find the factors of 6 and 11,
The factors of 6 are 1, 2, 3, and 6.
The factors of 11 are 1 and 11.
Here, the only shared factor of 6 and 11 is 1, and their GCD equals 1. Hence, 6 and 11 are coprime numbers.
Coprime numbers are two or more numbers whose greatest common factor is 1. Twin prime numbers are prime numbers that always have a difference of 2. The differences between coprime and twin numbers are listed below:
In computer science, mathematics, and engineering the concept of coprime numbers plays an important role. The real-world applications of coprime numbers are:
Understanding the concept of coprime numbers helps solve calculations related to prime and coprime numbers. Many students often make mistakes when they deal with coprime numbers. Here are some common mistakes and their helpful solutions to avoid these errors.
Check if 359 and 356 are coprime numbers.
359 and 356 are coprime numbers
To find whether 359 and 356 are coprime numbers, we must list the factors of both the given numbers.
359 is a prime number because its only factors are 1 and itself.
Factors of 356 include:
356 = 1 × 356
356 = 2 × 178
356 = 4 × 89
So, the factors of 356 are 1, 2, 4, 89, 178, and 356. The factors of 359 are 1 and 359.
Since the only common factor of 359 and 356 is 1, the numbers 356 and 359 are coprime.
Check whether 11 and 18 are coprime numbers.
Yes, 11 and 18 are coprime numbers.
11 is a prime number.
The factors of 11 are 1 and 11.
18 is a composite number. The factors of 18 are 1, 2, 3, 6, 9, and 18.
The common factor between 11 and 18 is only 1.
Therefore, the numbers 11 and 18 are coprime.
If 89 and 97 are coprime, what would be their GCF?
1
First, we must list the factors.
Factors of 89 = 1 and 89
Factors of 97 = 1 and 97
89 and 97 are prime numbers. So, the only common factor between 89 and 97 is 1.
Since both the numbers do not have other common factors except 1, their greatest common factor ( GCF) is 1.
GCF of 89 and 97 = 1
Are 12 and 56 coprime numbers?
12 and 56 are not coprime numbers.
To check whether 12 and 56 are coprime, we need to find the common factors beyond 1.
Factors of 12 = 1, 2, 3, 4, 6, and 12
Factors of 56 = 1, 2, 4, 7, 8, 14, 28, and 56
The common factors of 12 and 56 are 1, 2, and 4.
Since the numbers share common factors other than 1, 12 and 56 are not coprime.
Check if 17 and 18 are coprime numbers.
17 and 18 are coprime numbers.
17 and 18 are consecutive numbers. However, we can list the factors of both numbers.
Factors of 17 = 1 and 17.
17 is a prime number.
Factors of 18 = 1, 2, 3, 6, 9, and 18.
18 is a composite number.
The common factor between 17 and 18 is only 1.
Since the greatest common factor of both numbers is 1, they are coprime numbers.
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
: She loves to read number jokes and games.