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Last updated on July 22nd, 2025

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Coprime Numbers

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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.

Coprime Numbers for UAE Students
Professor Greenline from BrightChamps

What are 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:

 

 

  • Two prime numbers are always coprime. Each prime number has only two factors: 1 and itself. For example, (3, 7) is a coprime pair, because the only common factor is 1. 
  • If the GCD of two composite numbers is 1, they are coprime. For instance, (9, 4), both these numbers have multiple factors, but they do not share any common factors except 1. 
  • GCD or Greatest Common Divisor of a pair of coprime numbers is always 1.  
  •  For exactly two coprime numbers, the LCM is equal to their product. For example, (5, 9) is a coprime pair.
  • (5 × 9) = 45
  • Multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, …
  • Multiples of 9 are 9, 18, 27, 36, 45, 54, …
  • The smallest common number among both lists is 45.
  • The LCM of 5 and 9 = 45 
  • Any number and 1 can always form a pair of coprime numbers. For instance, (1, 3), (1, 7), (1, 10) are coprime pairs. 
  • Two even numbers cannot be a coprime pair because even numbers have 1 and 2 as common factors.
    For example, 
    Factors of 2 include 1 and 2.
    Factors of 44 include 1, 2, 4, 11, 22, and 44.
  • Two consecutive numbers are always coprime. For example, (3, 4), (4, 5), and (5, 6). 
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Difference Between Prime and Coprime Numbers

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). 

 

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How to Find Coprime Numbers?

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. 
 

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What are coprime and Twin Prime 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: 

 

 

  • Coprime numbers can be either composite or prime numbers. However, twin prime numbers only contain prime numbers.

 

  • The difference between two coprime numbers can be any value, but the difference between two twin prime numbers is 2. 

 

  • Not all coprime numbers form twin prime pairs, but every twin prime pair contains coprime numbers. 

 

  • Any number and 1 can form a coprime pair, whereas 1 is not part of any twin prime pair.  
     
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Real-life Applications of Coprime Numbers

In computer science, mathematics, and engineering the concept of coprime numbers plays an important role. The real-world applications of coprime numbers are: 

 

 

  • Coprime numbers are used to design encryption techniques that ensure the security of online transactions and communications. 

 

  • Professionals in the field of music and sound theory can use coprime numbers to create new tunes and synchronize the beats to avoid overlapping sound waves. 

 

  • In mathematics, coprime numbers play an essential role by aiding in simplifying fractions and laying the foundation for mathematical theorems. Understanding the concept helps students easily solve mathematical problems related to LCM and GCF. 

 

  • Managers can use coprime numbers to schedule the shifts of their employees and track their work schedules. For instance, the schedules of two employees who work shifts every 4 and 5 days will only repeat every 20 days. It helps the managers to distribute the work among the workers evenly and reduces the chances of overlapping. 
     
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Common Mistakes and How to Avoid Them on Coprime Numbers

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. 
 

Mistake 1

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Believing coprime numbers are prime numbers
 

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 Students should understand that coprime numbers can be a set of either a composite or a prime number. The main thing is that the coprime numbers should share only one factor, which is 1. If they mistakenly assume only prime numbers can be coprime, it can lead them to incorrect conclusions. For example, (8, 11) is a coprime pair as it consists of a composite and a prime number.  
 

Mistake 2

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Skipping all common factors
 

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 Kids should list all the possible factors of the given number to check whether the given number is coprime or not. If they forget to check all the common factors, they may skip any common factor and assume a non-coprime number is coprime. For example, (3, 9) are not coprime numbers because the factors of 3 include 1 and 3. Likewise, the factors of 9 include 1, 3, and 9.  
 

Mistake 3

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Forgetting that every consecutive number pair is coprime
 

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When dealing with coprime numbers, remember that every pair of consecutive positive numbers is coprime. All consecutive numbers share only one common factor, which is 1. For example, (1, 2), (2,3), (45, 46), (46, 47), and so on are examples of consecutive coprime numbers. 
 

Mistake 4

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Believing that even and odd numbers are always coprime 
 

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Students should remember to verify all the factors of the given numbers before finalizing the answers. Do not think that an even number and an odd number in a pair must be coprime. For example, (5,10) or (6, 9) are not coprime numbers because both pairs share more than one common factor. 
 

Mistake 5

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Ignoring every twin prime pair is a coprime
 

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 Keep in mind that twin prime pairs, that differ by 2, are always coprime. If students skip the concept of twin prime numbers, they will end up with incorrect conclusions. For example, (3, 5) and  (5, 7) are twin prime numbers that only share one factor, which is 1. 
 

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Solved Examples of Coprime Numbers

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Problem 1

Check if 359 and 356 are coprime numbers.

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 359 and 356 are coprime numbers
 

Explanation

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. 

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Problem 2

Check whether 11 and 18 are coprime numbers.

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Yes, 11 and 18 are coprime numbers. 
 

Explanation

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. 
 

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Problem 3

If 89 and 97 are coprime, what would be their GCF?

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1

Explanation

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
 

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Problem 4

Are 12 and 56 coprime numbers?

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12 and 56 are not coprime numbers. 

Explanation

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. 
 

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Problem 5

Check if 17 and 18 are coprime numbers.

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17 and 18 are coprime numbers.
 

Explanation

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.  
 

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FAQs on Coprime Numbers

1.What do you mean by Coprime numbers?

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2.Differentiate prime and coprime numbers.

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3.Are two even numbers always coprime?

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4.Can two prime numbers be Coprime?

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5.How can children in United Arab Emirates use numbers in everyday life to understand Coprime Numbers?

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6.What are some fun ways kids in United Arab Emirates can practice Coprime Numbers with numbers?

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7.What role do numbers and Coprime Numbers play in helping children in United Arab Emirates develop problem-solving skills?

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8.How can families in United Arab Emirates create number-rich environments to improve Coprime Numbers skills?

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Hiralee Lalitkumar Makwana

About the Author

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.

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Fun Fact

: She loves to read number jokes and games.

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