What is a co-prime number? | Definition, properties, and example

Co-prime numbers have no common factor other than 1. A set of co-prime numbers must consist of at least two numbers. For example, 4 and 7 have only 1 as their highest common factor. Co-prime numbers are also formed by two composite numbers, 4 and 9. Keep reading to learn what is a co-prime number and its definition and examples.

What is a co-prime number?

For example, x and y are two positive integers that are co-prime numbers if and only if they have one common factor, and thus HCF(x, y) = 1. In other words, co-prime numbers are a pair of numbers or integers with only one common factor, namely that their highest common factor (HCF) is 1. Co-prime numbers are also known as mutually prime numbers or relatively prime numbers. It is critical to have two numbers to form co-primes.

How to find co-prime numbers?

Consider a pair of numbers; if no positive integer other than 1 can divide them, the pair is co-prime.

Example 1- 21 and 22

For numbers 21 and 22-

Number	Factors
21	1, 3, 7
22	1, 2, 11

Here, 21 and 22 share only one factor, which is 1. As a result, the HCF is 1, and they are co-prime.

Example 2- 21 and 27

For numbers 21 and 27-

Number	Factors
21	1, 3, 7
27	1, 3, 9

In this case, 21 and 27 shares two factors, 1 and 3. They have an HCF of 3 and are not co-prime.

Co-prime numbers list

Here is the list of co-prime numbers-

Co-prime numbers with	Co-prime number pair
1	(1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1,7), (1,8), (1, 22), and so on
2	(2, 3), (2, 5), (2, 7), (2, 9), (2, 11), (2, 13), (2, 15), and so on
3	(3, 4), (3, 5), (3, 7), (3, 10), (3, 11), (3, 15), (3, 22), and so on
4	(4, 5), (4, 7), (4, 9), (4, 11), (4, 13), (4, 15), (4, 17), and so on
5	(5, 6), (5, 7), (5, 8), (5, 9), (5, 11), (5, 12), (5, 14), and so on

Properties

Co-prime numbers have six main properties, which are as follows-

• The universal co-prime number is 1. You can combine 1 with any number, and both will be co-prime. For example, (1,2), (1,3), (1,8), (1,15), (1,532), (1,2568).
• In a number sequence, the integer immediately following and preceding an integer are co-prime. Take, for example, the number 33; thus, the number next to 33, i.e., 34, is co-prime to 33; similarly, the number before 33, i.e., 32, is also co-prime to 33.
• Because 0 is a factor less number, it is not co-prime with any other number.
• Even numbers can never be coprime because they all share a common factor other than 1, which is 2.
• If we add two co-prime numbers and multiply them, the resulting numbers are also co-prime. For example, 3+4 = 7, and 3 x 4 = 12, the results 7 and 12 have only one common factor, and thus they are co-prime.
• An odd number and an even number are also co-prime. However, if the numbers have 0 and 5 at the ones or unit places, they are not co-prime because they have an HCF = 5.
• Prime numbers are co-prime numbers. For example,  (11,13); 11=1×11 & 13= 1×13

Co-prime and twin prime numbers

Co-prime numbers have an HCF of one. Twin prime numbers, on the other hand, are prime numbers that have a difference of two. For instance, 7 and 5 are twin prime numbers. The following points distinguish co-prime and twin prime numbers.

Co-prime number	Twin prime number
The difference between two co-primes can be any whole number.	The difference between two twin primes is always two.
Co-prime numbers can be either prime numbers or composite numbers.	Twin prime numbers are always prime numbers.
Co-prime numbers can be twin prime numbers or not.	All twin prime number sets are co-prime.
Any number can join 1 to form a coprime pair.	1 form a twin prime pair only with 3. For example, 1 and 3 as (3-1=2)

Co-prime numbers from 1 to 100

The are several pairs of co-primes ranging from 1 to 100. Among them are-

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

Furthermore, any number with the combination of 1 can be written as a coprime pair, such as (22, 1), (31, 1), (4, 1), (90, 1), (1, 100). Many coprime numbers from 1 to 100 are defined in this manner.

Key takeaways

• The co-prime number does not have to be a prime number. It can also be a composite number.
• Any two prime numbers given must be a Co-prime.
• Sets of two even numbers can never be co-prime.
• Any number can form a coprime pair with number one (one).
• Any two consecutive numbers are indeed co-prime numbers.
• Two even numbers can never be co-prime numbers
• Co-prime numbers need not be prime numbers. For instance, 12 and 35 are co-prime numbers, even though 12 and 35 are not prime numbers.

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FAQs

Q1. What is the co-prime of 60?

Answer- For example, 60 divides by two to yield 30, which divides by two to yield 15, which divides by three to yield five (another prime). Hence, 60 = 2 × 2 × 3 × 5.

Q2. What is the smallest co-prime number?

Answer- 2 is the smallest prime number. To be prime, a number must have only two aspects- 1 and the number itself.

Q3. How do you check if two numbers are coprime?

Answer- If the greatest common divisor of two numbers, X and Y, is 1, they are said to be Co-Prime or mutually prime.