Co prime numbers are two numbers that have no common factor other than 1. In other words, their HCF (highest common factor) is 1. For example, 8 and 15 are co-prime. The factors of 8 are 1, 2, 4 and 8. The factors of 15 are 1, 3, 5 and 15. The only factor they share is 1.
Co-prime numbers are also called relatively prime numbers or mutually prime numbers. All three names mean the same thing.
What are co-prime numbers?
Here is the simple definition: two numbers are co-prime if the only number that divides both of them exactly is 1.
The key point is that co-prime numbers do not need to be prime themselves. 8 is not prime and 15 is not prime, yet together they are co-prime. Being co-prime is about a pair of numbers, not a single number.
Co-prime numbers examples
| Pair | Factors | HCF | Co-prime? |
|---|---|---|---|
| 8 and 15 | 8: 1, 2, 4, 8 15: 1, 3, 5, 15 | 1 | Yes |
| 4 and 9 | 4: 1, 2, 4 9: 1, 3, 9 | 1 | Yes |
| 14 and 15 | 14: 1, 2, 7, 14 15: 1, 3, 5, 15 | 1 | Yes |
| 25 and 36 | 25: 1, 5, 25 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 | 1 | Yes |
| 12 and 18 | 12: 1, 2, 3, 4, 6, 12 18: 1, 2, 3, 6, 9, 18 | 6 | No |
| 7 and 21 | 7: 1, 7 21: 1, 3, 7, 21 | 7 | No |
More co-prime pairs from 1 to 50: (2, 9), (6, 35), (9, 28), (10, 21), (15, 28), (16, 27) and (35, 48).
How to find co-prime numbers
There are two easy methods.
Method 1: list the factors
- Write all the factors of the first number.
- Write all the factors of the second number.
- Find the common factors. If 1 is the only one, the numbers are co-prime.
Method 2: use prime factors
Break each number into prime factors. If no prime appears in both, they are co-prime.
Example: 15 = 3 × 5 and 28 = 2 × 2 × 7. No prime is shared, so 15 and 28 are co-prime. Example: 35 = 5 × 7 and 48 = 2 × 2 × 2 × 2 × 3. Again nothing is shared, so 35 and 48 are co-prime. The prime checker below shows the prime factors of any number, which makes this method quick.
Properties of co-prime numbers
- 1 is co-prime with every number. The only factor of 1 is 1.
- Two different prime numbers are always co-prime. For example, 3 and 5, or 11 and 13.
- Two consecutive numbers are always co-prime. For example, 8 and 9, 14 and 15, or 21 and 22. Any number dividing both would also divide their difference, which is 1.
- Two consecutive odd numbers are always co-prime. For example, 15 and 17, or 25 and 27.
- Two even numbers are never co-prime. They always share the factor 2. For example, the HCF of 4 and 6 is 2.
- The LCM of two co-prime numbers is their product. The LCM of 4 and 9 is 4 × 9 = 36.
Co-prime numbers vs twin prime numbers
These two ideas are easy to confuse. Twin prime numbers are pairs of primes that differ by exactly 2, such as 11 and 13. Co-prime numbers can be any two numbers with HCF 1, prime or not. So every twin prime pair is co-prime, but most co-prime pairs, like 8 and 15, are not twin primes.
Co-prime numbers from 1 to 100
There are a great many co-prime pairs between 1 and 100, so they are not listed one by one. Instead, use the rules above to spot them quickly:
- Any two different numbers from the prime numbers 1 to 100 list, such as 13 and 97.
- Any two numbers next to each other, such as 50 and 51 or 99 and 100.
- A prime and any number that is not its multiple, such as 7 and 30.
To learn more about the numbers that build all of these, read about prime numbers. Practise your factors with the multiplication tables, which make it easy to see common factors.
Questions people ask
What are co prime numbers?
Co prime numbers are two numbers whose only common factor is 1. Their HCF is 1. For example, 8 and 15 are co-prime even though neither of them is a prime number.
What is co prime numbers with examples?
Two numbers are co-prime when no number other than 1 divides both. Examples are (4, 9), (8, 15), (14, 15) and (25, 36). The pair (12, 18) is not co-prime because both divide by 6.
How to find co prime numbers?
List the factors of both numbers, or break them into prime factors. If 1 is the only common factor, or no prime is shared, the numbers are co-prime. For example, 15 = 3 × 5 and 28 = 2 × 2 × 7 share no prime.
Are co-prime numbers always prime?
No. Co-prime numbers do not need to be prime themselves. 8 and 9 are both composite, but they are co-prime because their only common factor is 1.
Are two consecutive numbers always co-prime?
Yes. Any number that divides both of them would also divide their difference, which is 1. So the only common factor is 1. Examples are 20 and 21, and 99 and 100.