Relatively Prime

Relatively 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.
   

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.
   

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.

Given below is a list of tips and tricks helpful in mastering relatively prime numbers.

 

• Check factors first: Identify all factors of each number to quickly spot relatively prime pairs.
  
   
• Use the GCF method: Two numbers are relatively prime if their greatest common factor is 1.
  
   
• Start with small numbers: Practice with smaller numbers to build confidence before moving to larger ones.
  
   
• Apply real-life examples: Use objects, dice, or cards to visualize and test relatively prime numbers.
  
   
• Turn practice into games: Create fun challenges or puzzles to make learning relatively prime numbers engaging.

In real life, co-prime numbers or relatively prime numbers are used in various fields of computer science, cryptography and many more. Some applications are as follows:

• 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.
  
   
• In calendar design and scheduling, relatively prime numbers are used to create repeating patterns that minimize overlaps, for example, in planning events or maintenance cycles that don’t coincide too often.