Co-Prime Numbers
You know that some numbers share common factors. For example, 12 and 18 share factors 1, 2, 3, and 6. But what about numbers like 8 and 15? Their only common factor is 1. Such pairs of numbers are called co-prime numbers.
Co-prime numbers are special because they do not "overlap" at all in terms of factors. This makes them very useful in simplifying fractions, finding LCM, and many other calculations.
This topic is part of the Playing with Numbers chapter in Class 6 NCERT Maths. Understanding co-prime numbers helps build a strong foundation for higher maths.
What is Co-Prime Numbers - Grade 6 Maths (Playing with Numbers)?
Definition: Two numbers are co-prime (also called relatively prime) if their only common factor is 1. In other words, the HCF of co-prime numbers is 1.
Examples of co-prime pairs:
- 2 and 3 → Common factors: {1} → Co-prime
- 8 and 15 → Factors of 8: {1, 2, 4, 8}. Factors of 15: {1, 3, 5, 15}. Common: {1} → Co-prime
- 7 and 11 → Both prime, different primes are always co-prime → Co-prime
Examples of pairs that are NOT co-prime:
- 6 and 9 → Common factors: {1, 3}. HCF = 3 (not 1) → NOT co-prime
- 12 and 18 → Common factors: {1, 2, 3, 6}. HCF = 6 → NOT co-prime
Important: Co-prime is a property of a pair of numbers, not of a single number. A number is not "co-prime" by itself — two numbers are co-prime to each other.
Co-Prime Numbers Formula
How to check if two numbers are co-prime:
Two numbers are co-prime if HCF = 1
Methods to check:
- Listing method: List all factors of both numbers. If the only common factor is 1, they are co-prime.
- Prime factorisation: Write the prime factorisation of both numbers. If they share NO common prime factor, they are co-prime.
- HCF method: Find the HCF. If HCF = 1, the numbers are co-prime.
Special property for co-prime numbers:
If a and b are co-prime, then LCM(a, b) = a × b
This is because co-prime numbers share no prime factors, so the LCM must include all factors of both.
Derivation and Proof
Understanding co-prime numbers through prime factorisation:
Example 1: Are 14 and 15 co-prime?
- 14 = 2 × 7
- 15 = 3 × 5
- Common prime factors? None. (14 has 2 and 7; 15 has 3 and 5. No overlap.)
- Therefore, HCF = 1 → 14 and 15 are co-prime.
Example 2: Are 14 and 21 co-prime?
- 14 = 2 × 7
- 21 = 3 × 7
- Common prime factor: 7
- HCF = 7 (not 1) → 14 and 21 are NOT co-prime.
Why consecutive numbers are always co-prime:
- Take any two consecutive numbers, say n and n+1.
- If a number d divides both n and (n+1), then d must divide their difference: (n+1) − n = 1.
- The only number that divides 1 is 1 itself.
- So HCF(n, n+1) = 1 → any two consecutive numbers are co-prime.
Types and Properties
Common patterns of co-prime numbers:
1. Any two different prime numbers are co-prime
- 2 and 3 → co-prime
- 7 and 13 → co-prime
- 11 and 29 → co-prime
- Reason: A prime has only factors 1 and itself. Two different primes cannot share a factor other than 1.
2. Consecutive numbers are always co-prime
- 4 and 5 → co-prime
- 20 and 21 → co-prime
- 100 and 101 → co-prime
3. 1 is co-prime with every number
- 1 and 5 → co-prime
- 1 and 100 → co-prime
- Reason: The only factor of 1 is 1, so HCF(1, n) = 1 for any n.
4. An even number and an odd number MAY or MAY NOT be co-prime
- 4 and 9 → co-prime (no common factor except 1)
- 6 and 9 → NOT co-prime (both divisible by 3)
5. A prime number and any number it does not divide are co-prime
- 7 and 10 → co-prime (7 does not divide 10)
- 7 and 14 → NOT co-prime (7 divides 14)
Solved Examples
Example 1: Example 1: Checking with Factor Lists
Problem: Are 12 and 25 co-prime?
Solution:
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 25: 1, 5, 25
- Common factors: {1}
- HCF = 1
Answer: Yes, 12 and 25 are co-prime.
Example 2: Example 2: Checking with Prime Factorisation
Problem: Are 18 and 35 co-prime?
Solution:
- 18 = 2 × 3²
- 35 = 5 × 7
- Common prime factors: None
- HCF = 1
Answer: Yes, 18 and 35 are co-prime.
Example 3: Example 3: Not Co-Prime
Problem: Are 24 and 36 co-prime?
Solution:
- 24 = 2³ × 3
- 36 = 2² × 3²
- Common prime factors: 2 and 3
- HCF = 2² × 3 = 12
Answer: No, 24 and 36 are NOT co-prime. HCF = 12.
Example 4: Example 4: Consecutive Numbers
Problem: Show that 15 and 16 are co-prime.
Solution:
- 15 = 3 × 5
- 16 = 2⁴
- Common prime factors: None
- HCF = 1
15 and 16 are consecutive numbers. Consecutive numbers are always co-prime.
Answer: Yes, 15 and 16 are co-prime.
Example 5: Example 5: Co-Prime but Not Prime
Problem: Give an example of two composite (non-prime) numbers that are co-prime.
Solution:
- Take 8 and 9.
- 8 = 2³ (composite)
- 9 = 3² (composite)
- Common prime factors: None
- HCF = 1
Answer: 8 and 9 are both composite but co-prime. Being co-prime does NOT mean being prime.
Example 6: Example 6: LCM of Co-Prime Numbers
Problem: Find the LCM of 8 and 15, given that they are co-prime.
Solution:
- Since 8 and 15 are co-prime (HCF = 1), their LCM = 8 × 15 = 120.
Verification:
- 120 ÷ 8 = 15 (no remainder) → 120 is a multiple of 8.
- 120 ÷ 15 = 8 (no remainder) → 120 is a multiple of 15.
Answer: LCM of 8 and 15 = 120.
Example 7: Example 7: Finding Co-Prime Pair from a Set
Problem: From the numbers 6, 10, 13, 21, find all co-prime pairs.
Solution:
- (6, 10): HCF = 2. Not co-prime.
- (6, 13): 6 = 2×3, 13 = prime. No common factor. Co-prime.
- (6, 21): HCF = 3. Not co-prime.
- (10, 13): 10 = 2×5, 13 = prime. Co-prime.
- (10, 21): 10 = 2×5, 21 = 3×7. Co-prime.
- (13, 21): Both have no common factor. Co-prime.
Answer: Co-prime pairs are: (6,13), (10,13), (10,21), (13,21).
Example 8: Example 8: Sharing Sweets Equally
Problem: Ravi has 16 laddoos and 21 jalebis. He wants to make gift boxes with the same number of laddoos in each box AND the same number of jalebis in each box, using all sweets. Can the number of boxes be more than 1?
Solution:
- The number of boxes must be a common factor of 16 and 21.
- 16 = 2⁴, 21 = 3 × 7. Common factors: {1}.
- 16 and 21 are co-prime.
Answer: No, only 1 box is possible (with all 16 laddoos and 21 jalebis). Since 16 and 21 are co-prime, they cannot be divided into more than 1 equal group simultaneously.
Example 9: Example 9: True or False
Problem: State True or False:
(a) 2 and 4 are co-prime.
(b) Two prime numbers are always co-prime.
(c) Two co-prime numbers are always prime.
Solution:
- (a) HCF(2, 4) = 2. Not 1. False.
- (b) Two different primes share no factor except 1. True.
- (c) Counter-example: 4 and 9 are co-prime but neither is prime. False.
Example 10: Example 10: Pizza Sharing Problem
Problem: A pizza is cut into 8 equal slices. Meera eats 3 slices. Can the fraction 3/8 be simplified further?
Solution:
- To simplify 3/8, we check if 3 and 8 are co-prime.
- 3 is prime. 8 = 2³. They share no common factor.
- HCF(3, 8) = 1. They are co-prime.
Answer: No, 3/8 is already in simplest form because 3 and 8 are co-prime.
Real-World Applications
Real-life uses of co-prime numbers:
- Simplifying fractions: A fraction a/b is in its simplest form when a and b are co-prime. For example, 3/7 is already simplified because 3 and 7 are co-prime (HCF = 1). But 6/8 is not in simplest form because HCF(6, 8) = 2; dividing by 2 gives 3/4.
- Finding LCM quickly: When two numbers are co-prime, their LCM is simply their product. LCM(5, 9) = 45 because 5 and 9 are co-prime. No prime factorisation needed.
- Gear mechanisms: In machines, gears with co-prime numbers of teeth wear more evenly because each tooth on one gear meshes with all teeth on the other before repeating.
- Computer science: Co-prime numbers are used in cryptography (RSA encryption) to create secure keys for online banking and messaging.
- Clock problems: The hour and minute hands of a clock overlap at intervals related to co-prime relationships between 12 and 1.
- Calendar patterns: The 7-day week and 365-day year are nearly co-prime (HCF = 1), which is why the same date falls on different days each year.
Key Points to Remember
- Two numbers are co-prime if their HCF is 1 (they share no common factor other than 1).
- Co-prime is also called relatively prime or mutually prime.
- Two different prime numbers are always co-prime.
- Two consecutive numbers are always co-prime (e.g., 14 and 15).
- 1 is co-prime with every number.
- Co-prime numbers need not be prime themselves (e.g., 8 and 9 are both composite but co-prime).
- If two numbers are co-prime, their LCM = their product.
- A fraction is in simplest form when the numerator and denominator are co-prime.
- Two even numbers are never co-prime (both divisible by 2).
- Co-prime is a relationship between a pair of numbers, not a property of a single number.
Practice Problems
- Check if 28 and 45 are co-prime using prime factorisation.
- Are 16 and 25 co-prime? Are 16 and 24 co-prime?
- Find three pairs of co-prime numbers from: 4, 9, 15, 22, 35.
- Prove that any two consecutive odd numbers (like 11 and 13) are co-prime.
- If two numbers are co-prime and their product is 77, what are the numbers?
- Is the fraction 14/33 in simplest form? Check if 14 and 33 are co-prime.
- Find the LCM of 11 and 13 using the fact that they are co-prime.
- Give an example of three numbers that are pairwise co-prime (every pair is co-prime).
Frequently Asked Questions
Q1. What does co-prime mean?
Two numbers are co-prime if their only common factor is 1. Their HCF is 1. For example, 8 and 15 are co-prime because the only number that divides both is 1.
Q2. Are co-prime numbers always prime?
No. Co-prime means the pair shares no common factor except 1. The numbers themselves can be composite. For example, 4 and 9 are both composite (4 = 2×2, 9 = 3×3), but they are co-prime because they share no common factor.
Q3. Are two even numbers ever co-prime?
No. Any two even numbers share the factor 2, so their HCF is at least 2. Since HCF is not 1, two even numbers can never be co-prime.
Q4. Can more than two numbers be co-prime?
Yes. We say numbers are pairwise co-prime if every pair among them is co-prime. For example, 3, 4, and 5 are pairwise co-prime: (3,4), (3,5), and (4,5) are each co-prime.
Q5. Why are consecutive numbers always co-prime?
If a number d divides both n and n+1, then d must divide their difference: (n+1) − n = 1. The only positive number that divides 1 is 1 itself. So the HCF of any two consecutive numbers is 1.
Q6. What is the LCM of two co-prime numbers?
When two numbers are co-prime, their LCM equals their product. LCM(a, b) = a × b when HCF(a, b) = 1. For example, LCM(7, 9) = 7 × 9 = 63.
Q7. How are co-prime numbers used in fractions?
A fraction is in its simplest (lowest) form when the numerator and denominator are co-prime. To simplify a fraction, divide both by their HCF until the HCF becomes 1. For example, 12/18 → divide by HCF (6) → 2/3 (2 and 3 are co-prime).
Q8. Is (0, 5) a co-prime pair?
No. Every non-zero number divides 0 (because 0 ÷ 5 = 0). So HCF(0, 5) = 5 (not 1). Therefore 0 and 5 are not co-prime.
Q9. Are twin primes always co-prime?
Yes. Twin primes are pairs of primes that differ by 2 (like 11 and 13). Since they are both prime and different, their only common factor is 1. So twin primes are always co-prime.
Q10. How do I quickly tell if two numbers are NOT co-prime?
If both are even → not co-prime. If both are divisible by 3 (check digit sums) → not co-prime. If both end in 0 or 5 → not co-prime (share factor 5). Any shared prime factor means they are not co-prime.
