# Co - Prime Numbers

What are numbers? Try recalling the different types of numbers that you have come across, natural numbers, whole numbers, integers, even numbers, odd numbers, prime numbers, composite numbers etc. Prime numbers are those numbers which have only 1 and the number itself as its factors. What are co- prime numbers? Is it same as prime numbers?

The answer to this question is No, as Co-prime number is a set of numbers or integers which have only 1 as their common factor. That is their highest common factor (HCF) will be 1. It is also known as relatively prime or mutually prime numbers.

It is prime important that there should be two numbers in order to form co-primes.

Consider a set of two numbers, if they have no positive integer that can divide both, other than 1, the pair of numbers is co-prime.

Ex 1:  21 and 22

21 and 22:

• The factors of 21 are 1, 3, 7 and 21.
• The factors of 22 are 1, 2, 11 and 22.

Here 21 and 22 have only one common factor that is 1.Hence their HCF is 1 and are co-prime.

Ex 2: 21 and 27

21 and 27:

• The factors of 21 are 1, 3, 7 and 21.
• The factors of 27 are 1, 2, 3, 9 and 27.

Here 21 and 27 have two common factors they are 1 and 3. HCF is 3 and they are not co-prime.

## PROPERTIES OF CO-PRIME NUMBERS

Some of the properties of co-prime numbers are as follows.

1. 1 is co-prime with every number.

2. Every prime number is co-prime to each other: As every prime number have only two factors 1 and the number itself, the only common factor of two prime numbers will be 1. For example, 2 and 3 are two prime numbers. Factors of 2 are 1, 2 and factors of 3 are 1, 3.  The only common factor is 1 and hence is co-prime.

3. Any two successive numbers/ integers are always co-prime: Take any consecutive number such as 2, 3 or 3, 4 or 5, 6 and so on; they have 1 as their HCF.

4. The sum of any two co-prime numbers are always co-prime with their product: 2 and 3 are co-prime and have 5 as their sum (2+3) and 6 as the product (2×3). Hence, 5 and 6 are co-prime to each other.

Example: Check whether 13 and 31 are co-prime.

Solution: 13 and 31 are two prime numbers; therefore, they are co-prime to each other. (Property 2)

The factors of 13 are 1, 13 and the factors of 31 are 1, 31. They have only 1 as their common factor.