Coprime Numbers
Coprime Numbers are a pair of numbers with just one factor in common, which is 1. To put it another way, a pair of relatively prime numbers is said to be co-prime if their highest common factor (HCF) is 1.
- 5 and 7 are Coprime as the common factor is only 1.
- 22 and 11 are NOT Coprime as there is a common factor 11 (apart from 1)
- 21 and 22 are Coprime as the only common factor is 1.
Table of Content
Properties of Coprime Number
Co-prime numbers' key properties are as follows :
- Two prime numbers are always coprime. They only share 1 as their common factor.
- Two Composite Numbers can also be Coprime if their GCD is 1. For example 9 and 4.
- A pair of coprime numbers' s GCD always 1.
- The result of multiplication of two co-primes is always their LCM.
- Every number and 1 makes a pair of co-prime numbers.
- Due to the fact that two even numbers always have 2 in common, they cannot both be co-prime numbers.
- The sum of two Coprime Numbers is also 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-primes from 1 to 100. We can write each number along with 1 as a co-prime pair.
List of Co Prime Numbers
Co prime pairs with numbers, starting from 1 are listed in the table below :
List of Co Prime Pairs | |
|---|---|
| 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),… |
How To Find 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.
For instance, 21 and 22
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.
For instance, 15 and 20
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.
Difference between Prime and Co Prime Numbers
While prime numbers can be part of a co-prime pair, co-prime numbers themselves do not need to be prime.
Let's compare prime numbers and co-prime numbers:
Prime vs. Co Prime Numbers | ||
|---|---|---|
| Aspect | Prime Numbers | Co-Prime Numbers |
| Definition | 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. |
| Divisibility | 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. |
| Examples | 2, 3, 5, 7, 11, 13, 17, etc. | (15, 28), (9, 16), (8, 21), etc. |
| Number of Numbers | Concerns a single number. | Involves a pair or set of numbers. |
| Nature of Numbers | Always integers greater than 1. | Can be any integers, including 1 and negative numbers. |
| Common Factor | The only common factor is the number itself and 1. | The only common factor is 1. |
| Special Properties | 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. |
Difference between Co Prime and Twin Prime Numbers
Co-prime numbers have their HCF as 1. Twin prime numbers, on the other hand, are prime numbers whose difference is always 2. For instance, 3 and 5 are twin prime numbers.
Let's make a comparison between co-prime and twin prime numbers are :
Co Primes vs. Twin Prime Numbers | ||
|---|---|---|
| Feature | Co Prime Numbers | Twin Prime Numbers |
| Nature of Numbers | Can be composite or prime | Always prime |
| Difference Between Numbers | Can be any number | Always 2 |
| Relationship to Each Other | May or may not be twin primes | Always co-prime |
| Special Pairs | Every number forms a co-prime pair with 1 | Only the pair (3, 1) is considered as twin primes. |
Also Check,
Solved Examples on Co Prime Numbers
Here we have solved some examples on co-prime numbers for your help.
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.
Co Prime Numbers Practice Exercise
Here are some practice questions on co prime numbers for you to solve :
1. If the product of two co prime numbers is 117, then find their LCM.
2. Are 17 and 68 co prime numbers? Justify.
3. Are 30 and 415 co prime numbers? State your reasons.
4. Are 216 and 215 co prime numbers?
5. Are 18 and 35 Coprime Pairs?
