Co-prime numbers are set of numbers which do not have any common factor between them other than one. This means that their highest common factor (HCF) is 1.
There should be at least two numbers in order to form co-primes.
Co-prime numbers are also called as relatively prime numbers.
By the definition of coprime numbers, if the given set of numbers have 1 as an only common factor then the given set of numbers will be coprime numbers.
Let us consider a set of two numbers
14 and 15
Factors of 14 are 1, 2, 7 and 14.
Factors of 15 are 1, 3, 5 and 15.
The common factor of 14 and 15 is only 1. So, 14 and 15 are coprime numbers.
11 and 17
Factors of 11 are 1 and 11.
Factors of 17 are 1 and 17.
Common factors of 11 and 17 are only 1. So, 11 and 17 are coprime numbers.
15 and 18
Factors of 15 are 1, 3, 5 and 15.
Factors of 18 are 1, 2, 3, 6, 9 and 18.
Common factors of 15 and 18 are 1 and 3. Since the given set of numbers have more than one factor as 3 other than factor as 1. So, 15 and 18 are not coprime numbers.
Some of the properties of co-prime numbers are as follows.
1 is coprime with every number.
Every prime number is co-prime to each other:
As every prime number has only two factors 1 and the number itself, the only common factor of two prime numbers will be 1. For example, 11 and 17 are two prime numbers. Factors of 11 are 1, 11 and factors of 17 are 1, 17. The only common factor is 1 and hence is co-prime.
Any two successive numbers are always coprime:
Consider any consecutive number such as 2, 3 or 3, 4 or 14 or 15 and so on; they have 1 as their HCF.
The sum of any two co-prime numbers is always coprime 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.
Solved Examples
Q1. Check coprime numbers from the given set of numbers
21 and 24
13 and 15
17 and 18
Ans.
a) 21 and 24 are not a coprime number because their common factors are 1and 3
b) 13 and 15 are coprime numbers because they are prime numbers.
c) 17 and 15 are coprime numbers because they are two successive numbers.
Q2. What is the difference between prime numbers and coprime numbers?
Ans.
A prime number is defined as a number which has no factor other than 1 and itself.
But, coprime numbers are considered in pairs and two numbers are coprime if they have a common factor as 1 only. Their HCF is 1.
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