Co Prime Numbers: Overview, Questions, Preparation

Numbers are classified depending on their properties in the theory of the number system. We will see one such category, known as co-prime numbers. But let us first comprehend the concept of prime numbers before we dive deep into it. Numbers of the prime apply to: 

Numbers of natural form < than 1. 

Which has just two divisors, 1 and the number itself. In other words, certain numbers can not be written as two smaller natural numbers products.  

Examples: 2, 3, 5, 7, 11, 13, etc. 

Now, with our key question in mind, what is the meaning of coprime numbers? 

'Co' indicates a 'couple'. So, let's assume c and d, we may describe a pair of integers as coprime or primes, or jointly prime or relatively prime if: 

1 is the only positive integer that splits both to give the remaining zero. 

The HCF of the two numbers is, in other terms, 1 

The numbers of Coprime are denoted as gcd(c, d)=1 or ( c, d)=1 or "c is prime to d." 

Source: NCERT

• Coprime numbers can not actually be prime numbers themselves. For example: (10,21), if taken separately, the numbers are not prime numbers, but they are co-prime if viewed as a pair. This concludes that coprime numbers may also be composite numbers. 
• By nature, prime numbers are co-prime. For example: (2,3), (5,11), (19,23) etc. There are just two variables for a prime number, i.e. 1, and the number itself. Therefore, a pair of two primes.
• Numbers can only have a common factor of 1.
• For all else, unity is regarded as co-prime. 
• The principle of co-prime is only applicable to natural numbers. 
• Co-prime often consists of two consecutive integers. They have 1 as their HCF, for instance: (2,3), (11,12), (99,100) and so on. 
• The sum of 2 co-prime numbers with the product of the same 2 co-prime numbers is always co-prime. The sum of 2 and 3 is 5, and the product of 2 and 3 is 6. Thus, co-prime numbers are 5 and 6. 
• So, g^2 and h^2 are both co-prime numbers if g and h are co-prime numbers. The squares of 5 and 6, i.e. 25 and 36, for example, are both co-prime for each other.

There are numerous pairs of 1 to 100 co-primes that obey the properties above. There are some of them: 

• (13 and 14) 
• (28 and 57) 
• (1 and 99) 
• (2 and 97) 
• (46 and 67)  
• (75 and 41) 

and so on as well. 

This concept is taught under the chapter Playing With Numbers in class 6. In this chapter, you will learn about the nature of various types of numbers and use the knowledge in solving questions. The weightage of this chapter is 8 marks. In class 10, there is use of this topic under chapter Number Theory.

• Check if 13 and 31 are co-prime numbers or not?

Solution: 13 and 31 are two prime numbers; hence they are co-prime to each other. 

• Check if 23 and 24 are co-prime numbers or not?

Solution:

For 23 and 24:

The factors of 23 are 1 and 23.

The factors of 24 are 1, 4, 6.

23 and 24 have only one common factor that is 1. They are co-prime.

• Are 81 and 16 co-prime numbers?

Solution:

Factors of 81 = 1, 3, 9, 27, 81

Factors of 16 = 1, 2, 4, 8, 16

Common factor= 1

They are co-prime numbers.

Q: Define co-prime numbers?

Q: Differentiate between prime and co-prime numbers?

Q: How can you find a number's co-prime?

Q: Verify whether 18 and 35 are co-prime or not?

Q: Is 1 for All Numbers co-prime?