You must have heard about coprime numbers in your classes while learning about prime numbers. Can you recall it? A set of numbers are said to be coprime numbers when the highest common factor between them is 1. The phrase, mutually prime numbers or relatively prime numbers can also be used in place of coprime numbers. It is mandatory that there should be a pair of numbers to form coprime. A few examples of coprime numbers are (71 and 79), (7,23), and so on. In this article, we will learn about coprime numbers in detail and try to find out how to identify and find them. But before that, it is very important that we learn about prime numbers too.
What Do You Mean by Coprime Numbers?
A pair of integers is said to be coprime if the only common divisor between them is 1. It is said that Eratosthenes formed an algorithm that helped in the calculation of prime numbers. The algorithm is popularly called Sieve of Eratosthenes. This further helped in the discovery of coprime numbers.
How do you Identify Coprime Numbers?
The easiest method which you can use to identify coprime numbers is by finding the factors of the numbers in consideration. Let us check our few examples to understand how.
Example: 7 and 13.
1 and 7 are the factors of 7 while 1 and 13 are the factors of 13.
Since 1 is the only factor in common between 7 and 13, they are coprime.
Example: 9 and 17.
1, 3, and 9 are the factors of 9 while 1 and 17 are the factors of 17.
Since 1 is the only factor in common between 9 and 17, they are coprimes.
Example: 6 and 15.
1, 2, 3, and 6 are the factors of 6 while 1, 3, 5, and 15 are the factors of 15.
We see here that except for the number 1, there is one more common factor between them which is 3. Thus they are not coprime.
Some Important Properties of Coprime Numbers
- A pair of even numbers can never be coprime of each other.
- The number which is coprime with every number is 1.
- A pair of prime numbers are coprime with each other.
- A set of two consecutive numbers is coprime with each other.
What Do You Mean by Prime Numbers?
From the set of natural numbers, those numbers which have two factors only, that is the numbers itself or the number 1 are called prime numbers. The first prime number on the number line is 2. It is very easy to identify prime numbers. Let us check out some examples of prime numbers, 61 is a prime number as it has only two factors, 61 and 1. However, 52 is not a prime number since it has factors more than one namely, 1,2,4,52, etc. An interesting thing to note about prime numbers is that they are always odd numbers except for the number 2. Now, one question might pop into your head. Is 1 a prime number? The answer is no. It is imperative that there must be two factors, 1 and the number itself to form a prime number which is not the case with the number 1.
Coprime Numbers Vs. Prime Numbers
Prime numbers are those numbers that have two factors only, 1 and the number itself. Some examples of prime numbers are 7, 13, 19, 41, 71, etc.
Coprime numbers are those numbers that have only one number as the common divisor which is 1. Some examples of coprime numbers are (7, 13), (11, 51), etc.
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