What Are Co Prime Numbers Definition Properties and Examples
The concept of co-prime numbers plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding co-prime numbers helps students solve HCF/LCM word problems, recognize number properties, and avoid common mistakes in competitive exams. Let's explore the world of co-prime numbers in detail, with simple explanations, stepwise checks, solved examples, and Vedantu's easy shortcuts for mastering the topic.
What Is Co-Prime Numbers?
Co-prime numbers (also called relatively prime numbers) are any two natural numbers that have no common factor other than 1. That means, if the highest common factor (HCF) or greatest common divisor (GCD) of two numbers is 1, they are co-prime. For example, 8 and 15 are co-prime since their only common factor is 1. You’ll find this concept applied in areas such as HCF/LCM calculations, rational numbers, and divisibility problems.
Key Formula for Co-Prime Numbers
Here’s the standard formula: If GCD(a, b) = 1, then a and b are co-prime numbers.
Co-Prime Numbers Examples
| Pair of Numbers | Are They Co-Prime? | Why? |
|---|---|---|
| 5, 7 | Yes | No common factor except 1 |
| 8, 15 | Yes | No common factor except 1 |
| 9, 12 | No | Common factor: 3 |
| 14, 15 | Yes | No common factor except 1 |
| 12, 18 | No | Common factor: 6 |
| 17, 19 | Yes | No common factor except 1 (both prime) |
How to Check if Two Numbers are Co-Prime?
Follow this easy step-by-step method to check co-primality:
- List all factors of both numbers.
- Check for any common factor other than 1.
- If there is no other common factor, the pair is co-prime.
- You can also use the HCF/GCD method — if HCF(a, b) = 1, they are co-prime.
Example: Are 18 and 25 co-prime?
1. Factors of 18: 1, 2, 3, 6, 9, 182. Factors of 25: 1, 5, 25
3. Common factors: only 1
4. Since no other common factor exists, 18 and 25 are co-prime.
Co-Prime Numbers from 1 to 100
Here are some popular co-prime pairs in the range 1 to 100, useful for quick school revision and worksheets:
| Co-Prime Pair | Reason |
|---|---|
| (1, 99) | 1 is co-prime with every number |
| (14, 15) | Consecutive numbers are always co-prime |
| (17, 60) | No common factor except 1 |
| (12, 25) | No common factor except 1 |
| (99, 100) | Consecutive numbers are always co-prime |
| (29, 31) | Both are prime, so automatically co-prime |
| (18, 35) | No common factor except 1 |
Prime Numbers vs. Co-Prime Numbers
| Prime Numbers | Co-Prime Numbers |
|---|---|
| A number that has only two factors: 1 and itself | A pair of numbers that have no common factor except 1 |
| E.g., 2, 3, 5, 7, 11 | E.g., (4, 9), (8, 15), (21, 22) |
| Prime is a property of a single number | Co-prime is a property of a pair (or group) of numbers |
| Every pair of primes is co-prime | But co-prime numbers need not be prime |
Step-by-Step Illustration
Let’s check if 16 and 27 are co-prime:
1. List factors of 16: 1, 2, 4, 8, 162. List factors of 27: 1, 3, 9, 27
3. Common factor: Only 1
4. Result: Since HCF(16, 27) = 1, they are co-prime numbers
Speed Trick or Vedic Shortcut
Here’s a quick shortcut: Two numbers are always co-prime if they are consecutive (like 35, 36), or if one is an odd number and the other is an even number not divisible by the same base factors. Use the HCF trick: Try dividing both numbers by 2, 3, 5, etc. If nothing matches except 1, they are co-prime.
Example Trick: To check if 51 and 80 are co-prime, check divisibility by 2, 3, 5, 7 (small primes). None match. Their HCF = 1. Answer: Co-prime!
Vedantu’s live classes often showcase more number hacks for school and Olympiad problems.
Try These Yourself
- Write five pairs of co-prime numbers between 1 and 50.
- Check if (44, 99) is a co-prime pair.
- Find all co-prime pairs from 28 to 34.
- Spot which among (18, 49), (21, 28), (40, 41) is not a co-prime pair.
Frequent Errors and Misunderstandings
- Confusing co-prime numbers with prime numbers (not all co-prime numbers are primes).
- Assuming two even numbers can be co-prime — except for (2, 1), two even numbers are never co-prime.
- Forgetting that 1 is co-prime with every number.
Relation to Other Maths Concepts
The idea of co-prime numbers connects closely with concepts like Highest Common Factor (HCF) and Lowest common multiple (LCM). Mastering this helps with Factors and Multiples and Prime Factorization—all of which are essential for JEE, NTSE, and school exams.
Classroom Tip
A great way to remember co-prime numbers is: If two numbers have "1" as their only common factor—they’re co-prime! Vedantu’s teachers use simple table hacks and divisibility games to build your co-prime skills in fun live sessions.
We explored co-prime numbers—from definition, formula, tables, mistakes, to connections with related topics. Keep practicing with Vedantu and grow confident in spotting and using co-prime numbers in Maths problems and real life.
Explore related topics: Prime Numbers | Factors and Multiples| Prime Factorization
FAQs on Co Prime Numbers and Their Meaning in Maths
1. What are co prime numbers?
Two numbers are co prime numbers if their HCF (Highest Common Factor) is 1. This means they have no common factor other than 1.
- They do not need to be prime numbers.
- They can be composite numbers.
- Example: 8 and 15 have HCF = 1, so they are co prime.
2. How do you find if two numbers are co prime?
Two numbers are co prime if their HCF equals 1. Follow these steps:
- Find all factors of both numbers or use the division method.
- Calculate their HCF (or GCD).
- If the HCF = 1, the numbers are co prime; otherwise, they are not.
- Example: HCF of 14 and 25 is 1, so they are co prime.
3. Can two composite numbers be co prime?
Yes, two composite numbers can be co prime if their HCF is 1. Composite numbers simply mean they have more than two factors.
- Example: 9 and 10 are composite.
- Factors of 9: 1, 3, 9.
- Factors of 10: 1, 2, 5, 10.
- Common factor is only 1, so they are co prime.
4. Are 1 and any number always co prime?
Yes, 1 is co prime with every number because the HCF of 1 and any number is 1. Since 1 has only one factor (1), it cannot share any other common factor.
- Example: HCF(1, 25) = 1.
- Therefore, 1 and 25 are co prime.
5. What is the difference between prime numbers and co prime numbers?
A prime number has exactly two factors, while co prime numbers are a pair of numbers whose HCF is 1. Key differences:
- Prime number: e.g., 7 (factors are 1 and 7).
- Co prime numbers: e.g., 8 and 15 (HCF = 1).
- Co primes can be composite; prime numbers refer to single numbers.
6. Are consecutive numbers always co prime?
Yes, consecutive numbers are always co prime because their HCF is always 1. Any two consecutive integers differ by 1.
- Example: 12 and 13 → HCF = 1.
- Example: 20 and 21 → HCF = 1.
- Therefore, every pair of consecutive numbers is co prime.
7. What is an example of co prime numbers?
An example of co prime numbers is 18 and 35 because their HCF is 1. Verification:
- Factors of 18: 1, 2, 3, 6, 9, 18.
- Factors of 35: 1, 5, 7, 35.
- Common factor is only 1.
- So, 18 and 35 are co prime.
8. What is the HCF of co prime numbers?
The HCF (Highest Common Factor) of co prime numbers is always 1. This is the defining property of co prime numbers.
- If HCF > 1, the numbers are not co prime.
- Example: HCF(16, 27) = 1 → co prime.
9. Can two even numbers be co prime?
No, two even numbers cannot be co prime because they always share at least the common factor 2. Since both are divisible by 2, their HCF is at least 2.
- Example: 4 and 6 → HCF = 2.
- Therefore, they are not co prime.
10. What is the formula related to co prime numbers?
If two numbers are co prime, then their LCM × HCF = Product of the numbers, and since HCF = 1, LCM = Product of the numbers. Formula:
- LCM(a, b) × HCF(a, b) = a × b
- If HCF = 1 → LCM = a × b
- Example: 5 and 7 → LCM = 35 (because 5 × 7 = 35).
