Pairs of numbers with only one common factor are known as co-prime numbers. They are also called mutually prime numbers or relatively prime numbers. It is essential to understand the concept of co-prime numbers to calculate the highest common factors in mathematics. In this article, you will learn everything about co-prime numbers, how to find them and their list and properties. Furthermore, we have provided a few solved example questions for practice.

Scroll down to find out what co-prime numbers are in mathematics.

By definition, any set of numbers or integers having only 1 as their common factor is known as co-prime numbers. A minimum of two numbers form a set of co-prime numbers in mathematics and are also referred to as mutually prime or relatively prime numbers. Moreover, the HCF or Highest Common Factor of these numbers is 1.

Let’s understand the concept of co-prime numbers through an example. If 1, 3, 7 and 21 are the factors of 21 and 1, 2, 11 and 22 are the factors of 22, then 1 is the only common factor among them. Thus, HCF (Highest Common Factor) of (21, 22) is 1, making them co-prime numbers.

To find whether two integers or numbers are co-prime numbers or not, we first find their factors. If they have only one common factor, usually 1, the pair of numbers are co-prime. This means their GCF (Greatest Common Factor) or HCF (Highest Common Factor) is one, 1. Also, it takes two numbers to form co-primes.

Similarly, in order to check if the given set of numbers is co-prime or not, consider their all-possible factors first and then find their HCF or GCF. Let us take an example to understand the tips to find co-prime numbers better:

Check whether 14 and 15 are co-prime numbers or not.

First, we will find factors of both, e.g.,

Factors of 14 = 1, 2, 7, 14

Factors of 15 = 1, 3, 5, 15

Thus, the HCF of 14 and 15 is only 1. Therefore, they are co-prime numbers.

Any pair of prime numbers with a difference of 2 are known as twin prime numbers. For instance, as we know, 3 and 5 are co-prime numbers, and their difference is also equal to 2 (5 – 3 = 2). Therefore, 3 and 5 are also twin prime numbers. Their HCF is also 1 only.

Following are a few points to help you understand the exact difference between co-prime and twin prime numbers.

Co-Prime Numbers | Twin Prime Numbers |

The difference between two co-prime numbers can be any number such as 1, 2, 6, 10 and so on. | The difference between two twin prime numbers is always 2 (two). |

They can be composite numbers. | They are always prime numbers. |

Pairs of co-prime numbers are not always twin prime numbers. | The pairs of twin prime numbers are always co-prime numbers. |

Following are a few properties, or you may call them tips and tricks to find out the pairs of co-prime numbers in mathematics. Give them all a read and find pairs of co-prime numbers by yourself:

- Pair numbers having 1 as their HCF or GCF are always co-prime numbers.
- Co-prime numbers are not always prime numbers.
- Two prime numbers are always co-prime numbers.
- Two consecutive or successive numbers or integers are always co-prime, such as 3, 4 or 5, 6, and so on.
- 1 always forms a co-prime pair with every number.
- Numbers having 5 and 0 at their unit place cannot make a co-prime pair. Such as 10 and 15.
- Add two co-prime numbers, and its sum will always co-prime with respective numbers’ product. For example, as we know, (2, 3) are co-prime numbers. Their sum is (2 + 3 = 5) and product is (2 x 3 = 6). Here, 5 and 6 are also co-prime to each other.
- All even numbers are divisible by 2. Therefore, two even numbers cannot make a pair of co-prime.

Here we have provided the most used co-prime numbers and their pairs in mathematics. Some of them are also twin prime numbers. Go through the table below to have a list of co-prime numbers pairs.

Co-Prime Number Pairs (1 to 50)

(1, 2) | (1, 11) | (1, 20) | (1, 30) | (1, 39) |

(1, 3) | (1, 12) | (1, 21) | (1, 31) | (1, 40) |

(1, 4) | (1, 13) | (1, 22) | (1, 32) | (1, 41) |

(1, 5) | (1, 14) | (1, 23) | (1, 33) | (1, 42) |

(1, 6) | (1, 15) | (1, 24) | (1, 34) | (1, 43) |

(1, 7) | (1, 16) | (1, 25) | (1, 35) | (1, 44) |

(1, 8) | (1, 17) | (1, 26) | (1, 36) | (1, 45) |

(1, 9) | (1, 18) | (1, 28) | (1, 37) | (1, 46) |

(1, 10) | (1, 19) | (1, 29) | (1, 38) | (1, 47) |

(2, 3) | (2, 25) | (2, 47) | (5, 21) | (1, 48) |

(2, 5) | (2, 27) | (2, 49) | (5, 23) | (1, 49) |

(2, 7) | (2, 29) | (3, 5) | (5, 27) | (1, 50) |

(2, 9) | (2, 31) | (3, 7) | (5, 29) | (5, 49) |

(2, 11) | (2, 33) | (3, 11) | (5, 31) | (7, 11) |

(2, 13) | (2, 35) | (5, 7) | (5, 33) | (7, 13) |

(2, 15) | (2, 37) | (5, 9) | (5, 37) | (7, 15) |

(2, 17) | (2, 39) | (5, 11) | (5, 39) | (7, 17) |

(2, 19) | (2, 41) | (5, 12) | (5, 41) | (7, 19) |

(2, 21) | (2, 43) | (5, 13) | (5, 43) | (7, 23) |

(2, 23) | (2, 45) | (5, 17) | (5, 47) | (7, 25) |

(7, 27) | (7, 31) | (5, 19) | (7, 9) | (7, 33) |

Several pairs of co-prime numbers such as (2, 97), (13, 14), (46, 67) and many more follow the properties of co-prime numbers. Furthermore, with the combination of 1, you can make a pair of co-prime numbers with any number or integer. For example (92, 1), (1, 100), and the list goes on.

Students struggling with math homework or any other concept must hire an experienced and qualified online or private math tutor. Numbers can sometimes be intimidating; that’s why we advise our students to go for extra help. Here are a few solved examples to help you understand the concept of finding co-prime numbers.

Solution:

First, check the common factors of both:

Factors of 18: 1, 2, 3, 6, 9, 18

Factors of 35: 1, 5, 7, 35

Here, the most common factor of 18 and 35 is 1.

Hence, 18 and 35 are co-prime numbers.

Solution:

Given that,

59 and 97 are co-prime numbers

Thus, their GCF or HCF will be 1.

Solution:

First, check the common factors of given numbers

Factors of 4: 1, 2 and 4

Factors of 8: 1, 2, 4 and 8

Common Factors between 4 and 8: 1 and 2

They have more than one common factor. Hence, 4 and 8 are not co-prime numbers.

To find the co-prime of a number, find a number that can not be divided by the factors of the other number. Also, their HCF or Highest Common Factor is always 1.

Prime Number: a number with no other factor except 1, and the number itself is known as a prime number. For example, 1, 3, 5, 7 and so on.

Co-Prime Numbers: A pair of numbers or two numbers having no common factor except 1 are known as co-prime numbers. For example, (1, 2), (3, 4), (5, 11) and so on.

According to the properties of co-prime numbers, 1 is the only co-prime of all other numbers or integers. Therefore, 1 is the co-prime of all numbers.

As 1 is the only common co-prime of all numbers. Therefore, the Highest Common Factor of two co-prime numbers is also 1, always.

In mathematics, co-prime numbers can be any two consecutive numbers. For example, 215 and 216. Both are co-prime numbers as well as consecutive numbers.

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