Common Factors
You already know how to find all the factors of a number. Now, what if you have two or more numbers and you want to find which factors they share? These shared factors are called common factors.
Imagine you have 12 red marbles and 18 blue marbles. You want to put them into bags so that each bag has the same number of red marbles AND the same number of blue marbles, with no marbles left over. The possible number of bags is determined by the common factors of 12 and 18: 1, 2, 3, and 6. The biggest number of bags you can make is 6 (with 2 red and 3 blue in each bag) — this number 6 is the Highest Common Factor (HCF) of 12 and 18.
In Class 6 NCERT Maths (Playing with Numbers), you will learn how to find common factors of two or more numbers, understand what the HCF is, learn three different methods to find it, and solve word problems about sharing, dividing, cutting, and arranging things equally.
Common factors are the stepping stone to understanding HCF, which is one of the most useful concepts in mathematics. HCF is used for simplifying fractions (dividing numerator and denominator by their HCF), distributing items equally, cutting materials into equal pieces, and many other practical situations that you encounter in daily life.
The concept of common factors also introduces an important idea: two numbers can be related through their shared structure. Numbers that share many common factors are "closely related" (like 12 and 18, which share factors 1, 2, 3, 6), while numbers that share only the factor 1 are called co-prime (like 8 and 15). Understanding this relationship helps in number theory, cryptography, and advanced mathematics.
What is Common Factors - Grade 6 Maths (Playing with Numbers)?
Definition: Common factors of two or more numbers are the factors that all the numbers share.
Finding common factors:
- List all factors of each number.
- Find the numbers that appear in every list.
Example:
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
- Common factors: 1, 2, 3, 6 (these appear in both lists)
Highest Common Factor (HCF):
- The largest of the common factors is called the Highest Common Factor (HCF), also known as Greatest Common Divisor (GCD).
- In the example above, HCF of 12 and 18 = 6.
Key facts:
- 1 is always a common factor of any two numbers.
- If one number is a factor of the other, the smaller number is the HCF. Example: HCF of 5 and 15 is 5.
- If the HCF of two numbers is 1, they are called co-prime numbers. Example: 8 and 15 are co-prime (common factor is only 1).
Common Factors Formula
Method 1: Listing Factors
- List all factors of each number.
- Circle or highlight the factors that are common to all lists.
- The largest common factor is the HCF.
Method 2: Prime Factorisation
- Find the prime factorisation of each number.
- Identify the common prime factors.
- Take each common prime with the smallest exponent.
- Multiply them together to get the HCF.
HCF = Product of common prime factors with smallest powers
Method 3: Division Method (for HCF)
- Divide the larger number by the smaller.
- If the remainder is 0, the divisor is the HCF.
- If the remainder is not 0, divide the previous divisor by the remainder.
- Repeat until the remainder is 0. The last divisor is the HCF.
Important relationship:
HCF × LCM = Product of the two numbers
HCF(a, b) × LCM(a, b) = a × b
Types and Properties
Type 1: Finding Common Factors by Listing
- List all factors of each number, then pick the common ones.
- Best for small numbers.
Type 2: Finding HCF Using Prime Factorisation
- Factorise both numbers into primes, then take common primes with smallest powers.
- Good for larger numbers.
Type 3: Finding Common Factors of Three Numbers
- List factors of all three numbers and find those common to all three lists.
- Example: Common factors of 12, 18, 24 are 1, 2, 3, 6.
Type 4: Identifying Co-prime Numbers
- Two numbers are co-prime if their only common factor is 1.
- Example: 9 and 14. Factors of 9: 1, 3, 9. Factors of 14: 1, 2, 7, 14. Common: only 1. Co-prime? Yes.
Type 5: Word Problems
- Sharing equally, cutting into equal pieces, arranging in groups.
- These problems use HCF to find the largest group size.
Type 6: Using Common Factors to Simplify Fractions
- Divide numerator and denominator by their HCF.
- Example: 18/24. HCF = 6. Simplified: 3/4.
Solved Examples
Example 1: Finding Common Factors by Listing
Problem: Find the common factors of 16 and 24.
Solution:
Given:
- Numbers: 16 and 24
Steps:
- Factors of 16: 1, 2, 4, 8, 16
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
- Common factors (in both lists): 1, 2, 4, 8
Answer: Common factors of 16 and 24 are 1, 2, 4, 8. HCF = 8.
Example 2: Finding HCF by Listing
Problem: Find the HCF of 20 and 30.
Solution:
Given:
- Numbers: 20 and 30
Steps:
- Factors of 20: 1, 2, 4, 5, 10, 20
- Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
- Common factors: 1, 2, 5, 10
- Highest common factor: 10
Answer: HCF of 20 and 30 = 10
Example 3: HCF Using Prime Factorisation
Problem: Find the HCF of 36 and 54 using prime factorisation.
Solution:
Given:
- Numbers: 36 and 54
Steps:
- 36 = 2² × 3²
- 54 = 2 × 3³
- Common primes: 2 and 3
- Smallest power of 2: 2¹
- Smallest power of 3: 3²
- HCF = 2 × 9 = 18
Answer: HCF of 36 and 54 = 18
Example 4: Common Factors of Three Numbers
Problem: Find the common factors of 12, 18, and 24.
Solution:
Given:
- Numbers: 12, 18, 24
Steps:
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
- Common to all three: 1, 2, 3, 6
Answer: Common factors are 1, 2, 3, 6. HCF = 6.
Example 5: Identifying Co-prime Numbers
Problem: Are 15 and 28 co-prime?
Solution:
Given:
- Numbers: 15 and 28
Steps:
- Factors of 15: 1, 3, 5, 15
- Factors of 28: 1, 2, 4, 7, 14, 28
- Common factor: only 1
- Since the only common factor is 1, they are co-prime.
Answer: Yes, 15 and 28 are co-prime.
Example 6: Simplifying a Fraction
Problem: Simplify 24/36 using common factors.
Solution:
Given:
- Fraction: 24/36
Steps:
- Find HCF of 24 and 36.
- 24 = 2³ × 3, 36 = 2² × 3²
- HCF = 2² × 3 = 12
- Divide both by HCF: 24 ÷ 12 = 2, 36 ÷ 12 = 3
Answer: 24/36 = 2/3
Example 7: Word Problem — Cutting Ribbons
Problem: Two ribbons are 24 cm and 36 cm long. They are to be cut into equal pieces of the longest possible length with no ribbon left over. How long is each piece?
Solution:
Given:
- Ribbon 1: 24 cm, Ribbon 2: 36 cm
Steps:
- The piece length must divide both 24 and 36 exactly.
- The longest such length is the HCF of 24 and 36.
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
- Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
- Common factors: 1, 2, 3, 4, 6, 12. HCF = 12.
Answer: Each piece should be 12 cm long. (24 ÷ 12 = 2 pieces, 36 ÷ 12 = 3 pieces.)
Example 8: Word Problem — Distributing Sweets
Problem: Reema has 40 ladoos and 60 barfis. She wants to pack them into boxes so that each box has the same number of ladoos and the same number of barfis, with none left over. What is the greatest number of boxes she can make?
Solution:
Given:
- Ladoos: 40, Barfis: 60
Steps:
- The number of boxes must divide both 40 and 60.
- Greatest number of boxes = HCF of 40 and 60.
- 40 = 2³ × 5, 60 = 2² × 3 × 5
- HCF = 2² × 5 = 20
- Each box: 40 ÷ 20 = 2 ladoos, 60 ÷ 20 = 3 barfis.
Answer: 20 boxes (each with 2 ladoos and 3 barfis).
Example 9: Word Problem — Tiling
Problem: A room is 16 m long and 12 m wide. What is the largest square tile that can be used to cover the floor without cutting any tile?
Solution:
Given:
- Room: 16 m × 12 m
Steps:
- The tile side must divide both 16 and 12.
- Largest tile side = HCF of 16 and 12.
- Factors of 16: 1, 2, 4, 8, 16
- Factors of 12: 1, 2, 3, 4, 6, 12
- Common factors: 1, 2, 4. HCF = 4.
- Tile size: 4 m × 4 m.
- Number of tiles: (16 ÷ 4) × (12 ÷ 4) = 4 × 3 = 12 tiles.
Answer: Largest tile = 4 m × 4 m. Total tiles needed: 12.
Example 10: HCF Using Division Method
Problem: Find the HCF of 48 and 18 using the division method.
Solution:
Given:
- Numbers: 48 and 18
Steps:
- Divide 48 by 18: 48 = 18 × 2 + 12 (remainder 12)
- Divide 18 by 12: 18 = 12 × 1 + 6 (remainder 6)
- Divide 12 by 6: 12 = 6 × 2 + 0 (remainder 0)
- Remainder is 0. The last divisor is 6.
Answer: HCF of 48 and 18 = 6
Real-World Applications
Real-world uses of common factors:
- Sharing Equally: If you have 24 apples and 36 oranges and want to distribute them equally into bags with no fruit left over, the number of bags must be a common factor of 24 and 36. Common factors are 1, 2, 3, 4, 6, 12. The maximum number of bags is the HCF = 12 (each bag gets 2 apples and 3 oranges). This kind of equal distribution problem appears in party planning, classroom activities, and gift distribution.
- Simplifying Fractions: To simplify 18/24, you need the HCF of 18 and 24. Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. HCF = 6. Divide both by 6: 18/24 = 3/4. This is the most common everyday use of common factors — every fraction simplification uses HCF, even if you do not explicitly think about it.
- Tiling and Flooring: To tile a rectangular room with the largest possible square tiles (no cutting), find the HCF of the room's length and width. For a room 18 m × 12 m, HCF = 6. Use 6 m × 6 m tiles. Number of tiles = (18/6) × (12/6) = 3 × 2 = 6 tiles. Architects and interior designers solve this problem regularly.
- Cutting Materials: A rope of 48 m and another of 60 m need to be cut into the longest possible equal pieces. The piece length must divide both 48 and 60. HCF(48, 60) = 12 m. The first rope gives 4 pieces and the second gives 5 pieces, totalling 9 pieces of 12 m each. This applies to cutting fabric, pipes, cables, and any material that comes in different lengths.
- Packaging and Manufacturing: A factory makes 120 chocolates and 180 biscuits daily. They want to pack them into identical gift boxes (same number of chocolates and biscuits in each). Maximum boxes = HCF(120, 180) = 60. Each box: 2 chocolates and 3 biscuits. Quality control, logistics, and packaging design all use HCF calculations.
- Music and Rhythm: Two musical patterns with 6 beats and 8 beats per cycle have HCF = 2. This means they share a common sub-rhythm of 2 beats. They align completely every LCM(6, 8) = 24 beats. Drummers and music producers use these ideas when layering different rhythmic patterns.
- Scheduling: If Bus A comes every 15 minutes and Bus B comes every 20 minutes, and both are at the stop at 8:00 AM, when will they next be at the stop together? LCM(15, 20) = 60 minutes = 1 hour later, at 9:00 AM. The HCF(15, 20) = 5 represents the common granularity of the schedules (both are multiples of 5 minutes).
Key Points to Remember
- Common factors are factors shared by two or more numbers. Example: Common factors of 12 and 18 are 1, 2, 3, 6.
- 1 is always a common factor of any set of numbers, because 1 divides every number.
- The Highest Common Factor (HCF) is the largest of the common factors. It is also called the Greatest Common Divisor (GCD). HCF of 12 and 18 = 6.
- If the HCF of two numbers is 1, they are called co-prime (or coprime). Example: 8 and 15 are co-prime (only common factor is 1). Co-prime numbers do not need to be prime themselves.
- Three methods to find HCF: listing factors (best for small numbers), prime factorisation (systematic, good for larger numbers), and division method (Euclidean algorithm — most efficient for very large numbers).
- Using prime factorisation: HCF = product of common prime factors with smallest exponents. HCF of 48 (2⁴ × 3) and 36 (2² × 3²) = 2² × 3 = 12.
- HCF × LCM = Product of the two numbers. This identity lets you find one if you know the other two. If HCF(12, 18) = 6 and 12 × 18 = 216, then LCM = 216 ÷ 6 = 36.
- HCF is used to simplify fractions: divide both numerator and denominator by their HCF. Example: 24/36 → HCF = 12 → 2/3.
- If one number is a factor of the other, the smaller number is the HCF. Example: HCF(5, 25) = 5, because 5 divides 25.
- The HCF of any number with itself is the number itself. HCF(7, 7) = 7.
- The HCF of any number with 1 is always 1.
- Common factors are always finite in number (unlike common multiples, which are infinite).
Practice Problems
- Find the common factors of 15 and 25.
- Find the HCF of 28 and 42 using the listing method.
- Find the HCF of 72 and 96 using prime factorisation.
- Are 21 and 32 co-prime? Show your working.
- Find the common factors of 18, 27, and 36.
- Simplify the fraction 48/60 using HCF.
- Two ropes are 45 m and 75 m long. Find the longest equal pieces they can be cut into.
- A farmer has 84 mangoes and 126 bananas. What is the greatest number of baskets he can make if each basket has the same number of mangoes and the same number of bananas?
Frequently Asked Questions
Q1. What are common factors?
Common factors of two or more numbers are the factors that all the numbers share. For example, factors of 8 are {1, 2, 4, 8} and factors of 12 are {1, 2, 3, 4, 6, 12}. Common factors are {1, 2, 4}.
Q2. What is the HCF?
HCF stands for Highest Common Factor. It is the largest factor that is common to all the given numbers. For 8 and 12, the common factors are {1, 2, 4}, so HCF = 4.
Q3. Is 1 always a common factor?
Yes. 1 divides every number exactly. So 1 is a factor of every number and therefore a common factor of any group of numbers.
Q4. What are co-prime numbers?
Two numbers are co-prime (or coprime) if their only common factor is 1, meaning HCF = 1. Examples: 7 and 10 (common factor: 1 only), 9 and 14 (common factor: 1 only). Note: co-prime numbers do not need to be prime themselves.
Q5. How is HCF different from LCM?
HCF is the largest number that divides all given numbers. LCM is the smallest number that is divisible by all given numbers. HCF ≤ both numbers ≤ LCM. Example: HCF of 6 and 8 is 2; LCM of 6 and 8 is 24.
Q6. How does HCF help simplify fractions?
To simplify a fraction, divide both the numerator and denominator by their HCF. Example: 36/48. HCF of 36 and 48 is 12. So 36/48 = (36 ÷ 12)/(48 ÷ 12) = 3/4.
Q7. Can the HCF be larger than either number?
No. The HCF can be at most equal to the smaller number (this happens when the smaller number divides the larger one). Example: HCF of 5 and 25 is 5.
Q8. How do you find common factors of three numbers?
List the factors of each number. Then find the numbers that appear in ALL three lists. These are the common factors. Alternatively, find the prime factorisation of each number and take common primes with the smallest exponents.
