102 LearnersLast updated on July 5th, 2025
Relatively Prime
Relative prime numbers are two numbers that have no common factors other than 1. Relative primes are used to identify patterns or simplify fractions. In this article, we will learn how to find relative prime numbers, their properties, and their applications.
What Are Relatively Prime Numbers?
Relative prime numbers are also known as co-prime or mutually prime numbers. If a and b are two numbers and their common factor is 1, then they are considered relatively prime. In this instance, the pair (a, b) is called a relatively prime pair. These numbers do not necessarily need to be prime; two composite numbers can also be relatively prime. For example, 9 and 10 are relatively prime, as the common factor is 1.
Here are some interesting fun facts about relatively prime numbers:
- The number 1 is relatively prime to every number, for example, 1 and 3, 1 and 8.
- Every pair of consecutive numbers, such as 6 and 7 or 11 and 12, is always relatively prime.
- The LCM of two prime numbers is always the product.
- No pair of even numbers is relatively prime because all even numbers have a common factor of 2.
What are the properties of relatively prime numbers?
Understanding the mathematical properties of relatively prime numbers helps us solve equations and simplify fractions. Here are a few properties to look at:
- Any two relatively prime numbers have a GCF (Greatest Common Factor) of 1. (For example, the GCF of 7 and 11 is 1)
- The LCM (The Least Common Multiple) of two relatively prime integers is always their product. For example, 4 × 9 = 36 is the LCM of 4 and 9, so 4 and 9 are relatively prime.
- The product of two relatively prime numbers is always relatively prime to their sum. For example, 3 × 5 = 15 is relatively prime to 3 + 5 = 8.
- All pairs of prime numbers are relatively prime because they do not have a common factor other than 1. For example, 13 and 23.
- Any pair of consecutive numbers is relatively prime. For example, 45 and 46.
How to Find Relatively Prime Numbers?
Now let’s learn how to find the relative prime numbers. To check if two numbers are relatively prime or not, we first find the Greatest Common Factor (GCF) of the given numbers:
- If the GCF of the numbers is 1, then the numbers are relatively prime.
- To find the GCF, we list all the factors of both numbers and identify which is the greatest number among the factors; it is the GCF.
Example 1: Determine if 13 and 20 are Relatively Prime
- Factors of 13: 1, 13
- Factors of 20: 1, 2, 4, 5, 10, 20
- Common factor: 1
As the GCF(13, 20) = 1. Therefore, the pair (13, 20) is relatively prime.
Example 2: Check if 18 and 24 are Relatively Prime
- Factors of 18: 1, 2, 3, 6, 9, 18
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
- Common factors: 1, 2, 3, 6
Since the GCF(18, 24) = 6, the pair (18, 24) is not relatively prime.
Real-Life Applications of Relatively Prime
In real life, co-prime numbers or relatively prime numbers are used in various fields of computer science, cryptography and many more. Few applications are as following -
- In traffic systems, the relative prime numbers are used to ensure safety by preventing congestion.
- In board games, relatively prime numbers are used to maintain fairness by avoiding predictable patterns.
- To secure financial transactions or online communications, the relative prime numbers are used.
- Musicians use these numbers to create unique beats and rhythms.
Common Mistakes and How to Avoid Them in Relatively Prime
Relatively prime numbers are a fundamental concept in math. However, students often make mistakes when finding or identifying them. Here are a few common mistakes and ways to avoid them:
Mistake 1
Mixing Up Relatively Prime and Prime
Some students often think that if the two numbers are relatively prime, then both numbers must be prime numbers. For example: 10 and 13. To avoid this, students should remember that the only common factor between relatively prime numbers is 1. The numbers don't need to be prime. For example, 10 and 13 where 10 is composite and 13 is prime, but the pair (10, 13) is relatively prime as the common factor is 1.
Mistake 2
Forgetting Special Cases
Students forget the special cases, such as pairs of consecutive numbers or a number paired with 1, such as (19, 20), (10, 1). To perform mental math, it is important to remember and understand the special cases.
Mistake 3
Missing Out Factors
Missing out on factors when listing them is one of the common mistakes students often make. It can result in errors when identifying relatively prime numbers. So, always verify whether the list of factors of the two numbers is correct or not.
Mistake 4
Confusing GCF with LCM
Students frequently confuse the concepts of GCF and LCM, so it is important to understand what LCM and GCF are. To avoid confusion, keep in mind that the LCM is the product of two relatively prime integers, and the GCF of two relatively prime numbers is always 1. Thus, for 5 and 11, the LCM is 5 × 11 = 55, and the HCF is 1.
Mistake 5
Relying Too Much on Even/ Odd Patterns
Some students might think that any even number paired with an odd number is always relatively prime. However, that’s not true. For example, the pair (10, 15) is not relatively prime because they share a common factor of 5.
Solution: Instead of relying on even or odd patterns, it's significant to always calculate the greatest common factor (GCF) to determine if two numbers are relatively prime.
Solved Examples of Relatively Prime
Problem 1
Determine if 9 and 24 are relatively prime.
9 and 24 are not relatively prime.
Explanation
We first find the factors of each number
Factors of 9: 1, 3, 9
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Now, identify the common factors.
Common factors: 1 and 3
Since 9 and 24 have a common factor greater than 1, i.e., 3, we can confirm that they are not relatively prime.
Problem 2
Determine if 17 and 31 are relatively prime.
17 and 31 are relatively prime.
Explanation
The first step is to find the factors of the given numbers:
Factors of 17: 1, 17 (since 17 is prime)
Factors of 31: 1, 31 (since 31 is prime)
Then, identify the common factors.
Common factor: Only 1
Now, find the GCF
GCF(17, 31) = 1
Since the GCF is 1, we confirm that 17 and 31 are relatively prime.
Problem 3
Check if 11 and 14 are relatively prime.
the GCF is 1, so we confirm that 11 and 14 are relatively prime.
Explanation
We first find the factors of the given numbers:
Factors of 11: 1, 11 (since 11 is prime)
Factors of 14: 1, 2, 7, 14
Then, identify the common factors.
Common factor: Only 1
GCF(11, 14) = 1
Here, the GCF is 1, so we confirm that 11 and 14 are relatively prime.
Problem 4
Check if 10 and 21 are relatively prime.
the GCF is 1, so we confirm that 10 and 21 are relatively prime.
Explanation
The first step is to find the factors of the given numbers separately:
Factors of 10: 1, 2, 5, 10
Factors of 21: 1, 3, 7, 21
Then, identify the common factors.
The only common factor is 1
Now, we find the GCF:
GCF(10, 21) = 1
Here, the GCF is 1, so we confirm that 10 and 21 are relatively prime.
Problem 5
Determine if 12 and 18 are relatively prime.
GCF(12, 18) = 6
Explanation
The first step is to list the factors separately to find the GCF:
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Now, list the Common factors of 12 and 18: 1, 2, 3, 6
Identify the largest common factor among them:
GCF(12, 18) = 6
FAQs on Relatively Prime
1.What do you mean by relatively prime numbers?
2.How are the LCM and GCF of relatively prime numbers connected?
3.Can we consider 1 a relative prime number?
4.What is the significance of relatively prime numbers?
5.How do we check whether two numbers are relatively prime?
6.How can children in Indonesia use numbers in everyday life to understand Relatively Prime?
7.What are some fun ways kids in Indonesia can practice Relatively Prime with numbers?
8.What role do numbers and Relatively Prime play in helping children in Indonesia develop problem-solving skills?
9.How can families in Indonesia create number-rich environments to improve Relatively Prime 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.
Fun Fact
: She loves to read number jokes and games.
