Have you observed that there are some number pairs which do not have any common factor apart from 1? These number pairs are known as coprime numbers. It is essential for you to know about coprime numbers while studying the concept of factors, GCD, fractions, and other mathematical topics.
Then what are coprime numbers? To put it simply, the two numbers can be said to be coprime if 1 happens to be their common factor. 8 and 15 are some coprime numbers since they do not have any common factor apart from 1. This article aims to discuss the concept of coprime numbers, examples, properties, and much more.
What Are Coprime Numbers?
Two numbers having 1 as their highest common factor are known as co-prime numbers.
Coprime numbers are alternatively known as relatively prime numbers.
Example:
Let us take two numbers 8 and 15.
Factors of 8 are 1, 2, 4, 8
Factors of 15 are 1, 3, 5, 15
HCF = 1
Therefore, 8 and 15 are coprime numbers.
It should be noted here that co-prime numbers do not need to be prime numbers always. Composite numbers too can be co-prime. For instance, 8 and 9 are both composite numbers but their HCF is 1 only.
Examples of Coprime Numbers
Two numbers having 1 as their highest common factor are known as co-prime numbers.
Coprime numbers are alternatively known as relatively prime numbers.
Example:
Let us take two numbers 8 and 15.
Factors of 8 are 1, 2, 4, 8
Factors of 15 are 1, 3, 5, 15
HCF = 1
Therefore, 8 and 15 are coprime numbers.
It should be noted here that co-prime numbers do not need to be prime numbers always. Composite numbers too can be co-prime. For instance, 8 and 9 are both composite numbers but their HCF is 1 only.
Here are some examples to make the concept of coprime numbers easier to understand. Two numbers are coprime when their Greatest Common Divisor (GCD) is 1.
| Number 1 | Number 2 | GCD | Coprime? |
|---|---|---|---|
| 8 | 15 | 1 | Yes |
| 9 | 10 | 1 | Yes |
| 14 | 25 | 1 | Yes |
| 7 | 20 | 1 | Yes |
| 12 | 18 | 6 | No |
| 15 | 25 | 5 | No |
For example, 14 has factors 1, 2, 7, and 14, while 25 has factors 1, 5, and 25. Their only common factor is 1, so they are coprime.
However, 12 and 18 have common factors such as 2, 3, and 6. Their GCD is 6, so they are not coprime numbers.
How to Find Coprime Numbers
Here are some important methods to find out that two numbers are coprime or not:
1. Check Their Factors
The factors of numbers must be checked to see what common factors they share.
For example: If we check 15 and 28.
Factors of 15: 1, 3, 5, 15
Factors of 28: 1, 2, 4, 7, 14, 28
There is just one common factor. Thus, 15 and 28 are coprimes.
2. GCD Method
GCD calculation is an easy technique to find out whether two numbers are coprime or not.
If:
GCD(a, b) = 1
Thus, the numbers are coprime.
For example:
GCD(9, 10) = 1
Therefore, 9 and 10 are coprimes.
However,
GCD(12, 18) = 6
Thus, 12 and 18 are not coprimes.
3. Prime Factorization
Prime factorization is also a good technique to check for coprime numbers.
For example:
18 = 2 x 3 x 3
25 = 5 x 5
Here, we see that the two numbers do not share any prime factor common between them. So, their GCD = 1
How to Find Coprime Numbers
Coprime numbers have some valuable properties.
GCD is 1
This is the most valuable property. If the numbers are coprime, then their GCD is 1.
Only One Common Positive Factor Exists
The only common positive factor for two coprime numbers is 1.
They Do Not Necessarily Need to Be Prime
Coprime numbers can be prime, composite, or both.
As an illustration, take 8 and 9; both of these are composite numbers, yet they are coprimes.
Consecutive Numbers Will Be Coprimes
Two consecutive positive numbers are always coprimes.
Some examples would be:
- 4 and 5
- 9 and 10
- 14 and 15
- 20 and 21
One is Prime and the Other is Composite
As an example, 7 is prime and 20 is composite. They are coprimes because their only common factor is 1.
Coprime Numbers vs Prime Numbers
Coprime numbers and prime numbers are not the same thing. Coprime numbers usually refer to a pair of numbers whose greatest common divisor (GCD) is 1, while a prime number has exactly two positive factors: 1 and itself.
| Coprime Numbers | Prime Numbers |
|---|---|
| Usually describe a pair of numbers | Describe one number |
| GCD of the pair is 1 | Has exactly two positive factors |
| Numbers can be composite | The number itself must be prime |
| Example: 8 and 15 | Example: 7 |
Why Are Coprime Numbers Important?
In mathematics, coprimes have various applications. Coprimes play an essential role when dealing with fractions.
For instance, 8/15 is the simplest form of a fraction as the numerator 8 and the denominator 15 are coprimes. The absence of common factors in the numbers beyond one makes it impossible for us to simplify the fraction any further.
Coprimes are also applicable in number theory, modular arithmetic, and other mathematics concepts that involve cryptography.
Conclusion
Two numbers having 1 as their highest common factor are known as co-prime numbers.
Coprime numbers are alternatively known as relatively prime numbers.
Example:
Let us take two numbers 8 and 15.
Factors of 8 are 1, 2, 4, 8
Factors of 15 are 1, 3, 5, 15
HCF = 1
Therefore, 8 and 15 are coprime numbers.
It should be noted here that co-prime numbers do not need to be prime numbers always. Composite numbers too can be co-prime. For instance, 8 and 9 are both composite numbers but their HCF is 1 only.
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