Co-prime numbers are pairs of numbers that have only one common factor, which is 1.
- There must be at least two numbers to form a set of co-prime numbers.
- A set of co-prime numbers is not always made up of prime numbers.
- The GCD (HCF) of a pair of co-prime numbers is always 1.
Properties of Coprime Number
- Two different prime numbers are always coprime because they share only 1 as their common factor.
- Two composite numbers can also be coprime if their GCD is 1. For example, 4 and 9.
- The GCD of a pair of coprime numbers is always 1.
- The LCM of two coprime numbers is equal to their product.
- Every number and 1 always form a pair of coprime numbers.
- Two even numbers cannot be coprime because they always have at least 2 as a common factor.
- The sum of two coprime numbers is always coprime with their product.
- Two consecutive numbers are always coprime. For example, 3 and 4, 4 and 5, etc.
Co Prime Numbers from 1 to 100
There are many pairs of co-prime numbers between 1 and 100. Any number paired with 1 forms a co-prime pair.
| Co-prime with | Co-prime numbers pairs |
|---|---|
| 1 | (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),…, (1, 20),…. |
| 2 | (2, 3), (2, 5), (2, 7), (2, 9),…, (2, 15),….. |
| 3 | (3, 4), (3, 5), (3, 7), (3, 10), (3, 11),…., (3, 20),… |
| 4 | (4, 5), (4, 7), (4, 9), (4, 11), (4, 13), (4, 15),…. |
| 5 | (5, 6), (5, 7), (5, 8), (5, 9), (5, 11), (5, 12),… |
Check for Co-Prime Numbers
A pair of integers is co-prime if there is no other positive integer (other than 1) that can divide them both.
Example 1:
21 and 22:
- 1, 3, 7, and 21 are the factors of 21.
- 1, 2, 11, and 22 are the factors of 22.
Here, 21 and 22 only share a single factor, which is 1. Since they are co-prime, their HCF equals 1.
Example 2:
15 and 20:
- 1, 3, 5, and 15 are the factors of 21.
- 1, 4, 5, and 20 are the factors of 27.
Here, the numbers 1 and 5 are two factors that 21 and 27 both share. They are not co-prime and HCF is 5.
Prime vs Co Prime Numbers
| Prime Numbers | Co-Prime Numbers |
|---|---|
| A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. | Co-prime numbers are two or more numbers that have no common positive divisor other than 1. |
| A prime number can be divided evenly only by 1 and itself. | Co-prime numbers can be divided evenly only by 1 when considered together. |
| 2, 3, 5, 7, 11, 13, 17, etc. | (15, 28), (9, 16), (8, 21), etc. |
| Concerns a single number. | Involves a pair or set of numbers. |
| Always integers greater than 1. | Can be any integers, including 1 and negative numbers. |
| The only common factor is the number itself and 1. | The only common factor is 1. |
| All prime numbers are odd, except for 2, which is the only even prime number. | Co-prime numbers do not need to be prime; for example, 8 (which is not prime) and 9 are co-prime. |
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Solved Examples
Example 1: Check if 259 and 256 are co-prime numbers.
Solution:
Given numbers are 259 and 256.
1,2,4,8,16,32,64,128 and 256 are the factors  of 256.
1,7,37 and 259 are the factors of 259.
They only have 1 as their common factor hence they are co-prime numbers.
Example 2: Check whether 28 and 21 are co-prime numbers.
Solution:
Given numbers are 28 and 21.
1,3,7 and 21 are the factors of 21.
1,2,4,7 and 28 are the factors of 28.
21 and 28 have 1,7 as their common factors. Since their highest common factor is 7.Hence, they are not co-prime numbers.
Unsolved Numbers on Co-Prime Numbers
Question 1: Check whether 45 and 64 are co-prime numbers.
Question 2: Check whether 32 and 27 are co-prime numbers.
Question 3: Check whether 18 and 35 are co-prime numbers.
Question 4: Check whether 49 and 50 are co-prime numbers.
Question 5: Check whether 26 and 39 are co-prime numbers.
Question 6: Check whether 81 and 125 are co-prime numbers.