Co-prime Numbers
Co-prime numbers (also called relatively prime numbers or mutually prime numbers) are a pair of numbers whose only common factor is 1. In other words, their HCF is 1.
This concept is important in Class 5 because it connects factors, HCF, and fractions. Understanding co-prime numbers helps students simplify fractions to their lowest terms and solve problems involving LCM efficiently.
A common misconception is that co-prime numbers must be prime numbers themselves. This is not true. For example, 8 and 15 are both composite numbers, but they are co-prime because they share no common factor other than 1. The term "co-prime" describes the relationship between two numbers, not a property of any single number.
Knowing whether two numbers are co-prime has practical value. When the numerator and denominator of a fraction are co-prime, the fraction is already in its simplest form. When two numbers are co-prime, their LCM is simply their product — no extra calculation needed.
What is Co-prime Numbers - Class 5 Maths (Factors and Multiples)?
Two numbers are co-prime (or coprime) if their Highest Common Factor (HCF) is 1. This means they share no common factor other than 1.
Two numbers a and b are co-prime if HCF(a, b) = 1
Important: Co-prime is a property of a pair of numbers, not of a single number. A number by itself is neither co-prime nor not co-prime — it depends on what it is paired with.
Example: 8 and 15 are co-prime because:
- Factors of 8: 1, 2, 4, 8
- Factors of 15: 1, 3, 5, 15
- Common factor: only 1
- HCF(8, 15) = 1
Types and Properties
Common pairs of co-prime numbers:
- Any two consecutive numbers are always co-prime. Example: (7, 8), (14, 15), (99, 100).
- Any two different prime numbers are always co-prime. Example: (3, 7), (11, 13), (5, 23).
- 1 and any number are always co-prime. Example: (1, 50), (1, 999).
- A prime number and a number not divisible by it are co-prime. Example: (7, 20) since 20 is not divisible by 7.
Pairs that are NOT co-prime:
- Any two even numbers share the factor 2. Example: (4, 6) has HCF = 2, not co-prime.
- Numbers where one is a multiple of the other. Example: (5, 25) has HCF = 5, not co-prime.
Solved Examples
Example 1: Example 1: Basic Check
Problem: Are 12 and 35 co-prime?
Solution:
Step 1: Factors of 12 = 1, 2, 3, 4, 6, 12
Step 2: Factors of 35 = 1, 5, 7, 35
Step 3: Common factors = only 1
Step 4: HCF(12, 35) = 1
Answer: Yes, 12 and 35 are co-prime.
Example 2: Example 2: Not Co-prime
Problem: Are 18 and 24 co-prime?
Solution:
Step 1: Factors of 18 = 1, 2, 3, 6, 9, 18
Step 2: Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24
Step 3: Common factors = 1, 2, 3, 6
Step 4: HCF(18, 24) = 6 (not 1)
Answer: No, 18 and 24 are not co-prime because they share factors 2, 3, and 6.
Example 3: Example 3: Consecutive Numbers
Problem: Show that 15 and 16 are co-prime.
Solution:
Step 1: 15 = 3 × 5
Step 2: 16 = 2 × 2 × 2 × 2
Step 3: No common prime factor.
Step 4: HCF(15, 16) = 1
Answer: 15 and 16 are co-prime. Any two consecutive numbers are always co-prime.
Example 4: Example 4: Two Prime Numbers
Problem: Are 11 and 29 co-prime?
Solution:
Step 1: 11 is a prime number (factors: 1, 11).
Step 2: 29 is a prime number (factors: 1, 29).
Step 3: Two different prime numbers share only the factor 1.
Answer: Yes, 11 and 29 are co-prime.
Example 5: Example 5: Using Prime Factorisation
Problem: Are 28 and 45 co-prime? Use prime factorisation.
Solution:
Step 1: Prime factorisation of 28 = 2 × 2 × 7
Step 2: Prime factorisation of 45 = 3 × 3 × 5
Step 3: No common prime factor between them.
Step 4: HCF(28, 45) = 1
Answer: Yes, 28 and 45 are co-prime.
Example 6: Example 6: Co-prime with 1
Problem: Is 1 co-prime with 84?
Solution:
Step 1: Factors of 1 = {1}
Step 2: Factors of 84 include 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
Step 3: Common factor = only 1
Step 4: HCF(1, 84) = 1
Answer: Yes, 1 is co-prime with every number.
Example 7: Example 7: Two Even Numbers
Problem: Are 14 and 22 co-prime?
Solution:
Step 1: Both 14 and 22 are even numbers.
Step 2: Any two even numbers share at least the factor 2.
Step 3: HCF(14, 22) = 2 (not 1).
Answer: No, 14 and 22 are not co-prime. Two even numbers can never be co-prime.
Example 8: Example 8: Application with Fractions
Problem: Is the fraction 8/15 already in its simplest form?
Solution:
Step 1: A fraction is in simplest form when numerator and denominator are co-prime.
Step 2: Check if HCF(8, 15) = 1.
Step 3: 8 = 2 × 2 × 2; 15 = 3 × 5. No common factor.
Step 4: HCF(8, 15) = 1. They are co-prime.
Answer: Yes, 8/15 is already in simplest form.
Example 9: Example 9: LCM of Co-prime Numbers
Problem: Find the LCM of 9 and 14, given that they are co-prime.
Solution:
Step 1: Verify: 9 = 3 × 3; 14 = 2 × 7. No common factor. They are co-prime.
Step 2: For co-prime numbers: LCM = product of the numbers.
Step 3: LCM(9, 14) = 9 × 14 = 126.
Answer: LCM of 9 and 14 = 126.
Example 10: Example 10: Finding a Co-prime Partner
Problem: Find a number between 10 and 20 that is co-prime with 18.
Solution:
Step 1: 18 = 2 × 3 × 3. We need a number with no factor of 2 or 3.
Step 2: Check numbers 11 to 19:
- 11 (prime, not 2 or 3) — co-prime with 18
- 13 (prime, not 2 or 3) — co-prime with 18
- 17 (prime, not 2 or 3) — co-prime with 18
- 19 (prime, not 2 or 3) — co-prime with 18
Answer: 11, 13, 17, and 19 are all co-prime with 18.
Real-World Applications
Why co-prime numbers matter:
- Simplifying fractions: A fraction is in its lowest form when the numerator and denominator are co-prime.
- Finding LCM quickly: If two numbers are co-prime, their LCM equals their product. This saves time.
- Number puzzles: Many problems in competitive maths use co-prime properties.
- Gear ratios: In machines, co-prime gear teeth ensure even wear on all teeth.
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 a property of a pair, not a single number.
- Any two consecutive numbers (e.g., 8 and 9) are always co-prime.
- Any two different prime numbers (e.g., 5 and 13) are always co-prime.
- 1 is co-prime with every number.
- Two even numbers are never co-prime (they always share factor 2).
- If two numbers are co-prime, their LCM = their product.
- A fraction is in simplest form when numerator and denominator are co-prime.
Practice Problems
- Are 25 and 36 co-prime? Find their HCF to check.
- List all numbers between 1 and 20 that are co-prime with 12.
- Show that any two consecutive numbers are co-prime by checking (20, 21).
- Priya says 9 and 27 are co-prime because 9 is odd and 27 is odd. Is she correct? Explain.
- If two numbers are co-prime and their product is 221, find the LCM of the two numbers.
- Is 16/45 in its simplest form? Use the co-prime test.
- Find three pairs of co-prime numbers from: 6, 11, 14, 25, 33.
- Aman picks two cards: 15 and 28. Are these numbers co-prime? Can he simplify the fraction 15/28?
Frequently Asked Questions
Q1. What does co-prime mean?
Two numbers are co-prime (or coprime) if their only common factor is 1, meaning their HCF equals 1. For example, 8 and 9 are co-prime because no number other than 1 divides both.
Q2. Are co-prime numbers the same as prime numbers?
No. Prime numbers have exactly two factors (1 and themselves). Co-prime is a relationship between two numbers — it means they share no common factor other than 1. Two composite numbers like 8 and 15 can be co-prime.
Q3. Can two even numbers be co-prime?
No. Every even number has 2 as a factor. So any two even numbers share the factor 2, making their HCF at least 2. They cannot be co-prime.
Q4. Are all consecutive numbers co-prime?
Yes. Any two consecutive numbers (like 50 and 51) are always co-prime. If a number d divides both n and n+1, then d must also divide their difference, which is 1. So d can only be 1.
Q5. How do co-prime numbers help with fractions?
A fraction is in its simplest (lowest) form when the numerator and denominator are co-prime. For example, 7/12 is in simplest form because HCF(7, 12) = 1.
Q6. What is the LCM of two co-prime numbers?
If two numbers are co-prime, their LCM is simply their product. For example, since HCF(5, 8) = 1, LCM(5, 8) = 5 × 8 = 40.
Q7. Can three numbers be co-prime?
Yes. Three numbers are called pairwise co-prime if every pair among them is co-prime. For example, 4, 9, and 25 are pairwise co-prime because HCF(4,9) = 1, HCF(4,25) = 1, and HCF(9,25) = 1.
Q8. Is 1 co-prime with every number?
Yes. The only factor of 1 is 1 itself, so HCF(1, n) = 1 for any number n. Therefore, 1 is co-prime with every number.
Q9. Is this topic in the NCERT Class 5 syllabus?
Yes. Co-prime numbers are introduced in the Factors and Multiples chapter of NCERT Class 5 Mathematics. Students learn to identify co-prime pairs and apply the concept to simplify fractions.
