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Finding Common Ground

NCERT Class 7 · Mathematics Based on NCERT Class 7 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter explores the concepts of Highest Common Factor (HCF) and Lowest Common Multiple (LCM) using real-life scenarios like tiling floors, packing rice, and making torans. It builds on the idea of factors and multiples from earlier grades and introduces prime factorisation as a systematic method to find HCF and LCM. The big idea a student must walk away with is that HCF and LCM are powerful tools to solve problems involving grouping, sharing, and finding common ground between numbers, and that prime factorisation makes these calculations efficient and reliable.
Key Concepts (14)
Highest Common Factor (HCF)
The highest (or greatest) of the common factors of two or more numbers. It is also known as the Greatest Common Divisor (GCD).
e.g. For a room of 12 ft by 16 ft, the largest square tile that fits exactly has side 4 ft, which is the HCF of 12 and 16.
Greatest Common Divisor (GCD)
Another name for the Highest Common Factor (HCF).
e.g. The GCD of 12 and 16 is 4.
Common Factors
Factors that are shared by two or more numbers.
e.g. The common factors of 12 and 16 are 1, 2, and 4.
Lowest Common Multiple (LCM)
The lowest (or smallest or least) of the common multiples of two or more given numbers.
e.g. The shortest toran that can be made using 6 cm strips and 8 cm strips is 24 cm, which is the LCM of 6 and 8.
Common Multiples
Multiples that are shared by two or more numbers.
e.g. Common multiples of 6 and 8 include 24, 48, 72, etc.
Prime Factorisation
Writing a number as a product of its prime factors.
e.g. 90 = 3 × 3 × 2 × 5.
Prime Number
A number greater than 1 that has only 1 and the number itself as its factors.
e.g. 2, 3, 5, 7, 11 are prime numbers.
Composite Number
A number that has more than two factors.
e.g. 12 is composite because its factors are 1, 2, 3, 4, 6, 12.
Factor
A number that divides another number exactly, leaving no remainder.
e.g. 4 is a factor of 12 because 12 ÷ 4 = 3.
Multiple
The product of a given number and any whole number.
e.g. Multiples of 7 are 7, 14, 21, 28, ...
Conjecture
A statement or claim made without proof or verification.
e.g. Anshu claimed 'The larger a number is, the longer its prime factorisation will be.' This was disproved by a counterexample (96 and 121).
Generalisation
A statement describing a pattern or property that holds in all possible cases.
e.g. If one number is a factor of another, then the HCF is the smaller number.
Counterexample
An example that proves a conjecture is false.
e.g. 96 = 2×2×2×2×2×3 (6 primes) and 121 = 11×11 (2 primes) disproves Anshu's conjecture.
Subpart
A product of a selection of prime factors from the prime factorisation of a number.
e.g. From 840 = 2×2×2×3×5×7, the subpart 2×2×7 = 28 is a factor of 840.
Formulas & Equations (3)
HCF × LCM = Product of the two numbers
Reads as: HCF times LCM equals the product of the two numbers
Variables: HCF = Highest Common Factor, LCM = Lowest Common Multiple
Use when: When you know two numbers and either their HCF or LCM, and need to find the other.
⚠ Students often forget that this formula only works for two numbers, not three or more.
HCF = Product of common primes with minimum occurrences
Reads as: HCF equals the product of common primes, each raised to the smallest exponent found in the factorisations
Variables: Common primes = primes appearing in all numbers; minimum occurrences = smallest count of that prime across the numbers
Use when: When finding HCF using prime factorisation without listing all factors.
⚠ Students include a prime that appears in only one number, or take the maximum instead of minimum occurrences.
LCM = Product of all primes with maximum occurrences
Reads as: LCM equals the product of all primes, each raised to the largest exponent found in any of the factorisations
Variables: All primes = every prime that appears in any number; maximum occurrences = largest count of that prime across the numbers
Use when: When finding LCM using prime factorisation without listing all multiples.
⚠ Students take the minimum instead of maximum occurrences, or forget to include a prime that appears in only one number.
How It Works
Finding HCF by Listing Factors
  1. List all factors of each number.
  2. Identify the common factors (factors that appear in all lists).
  3. The highest among the common factors is the HCF.
💡 This method becomes cumbersome for large numbers and you may miss some factors.
Finding LCM by Listing Multiples
  1. List the first several multiples of each number.
  2. Identify the common multiples (multiples that appear in all lists).
  3. The smallest common multiple is the LCM.
💡 This method is tedious for large numbers; prime factorisation is more reliable.
Prime Factorisation by Division Method
  1. Write the number and draw a vertical line to its left.
  2. Divide the number by the smallest prime that divides it exactly (e.g., 2, 3, 5, 7...).
  3. Write the quotient below the number.
  4. Repeat the process with the quotient until you reach 1.
  5. The prime factors are the divisors used, listed in order.
💡 Always start with the smallest prime (2) and move to larger primes only when the number is no longer divisible by 2.
Finding HCF using Prime Factorisation
  1. Find the prime factorisation of each number.
  2. Identify the common primes (primes that appear in all factorisations).
  3. For each common prime, take the minimum number of times it appears across the factorisations.
  4. Multiply these selected primes together to get the HCF.
💡 If there is no common prime factor, the HCF is 1.
Finding LCM using Prime Factorisation
  1. Find the prime factorisation of each number.
  2. List all primes that appear in any factorisation.
  3. For each prime, take the maximum number of times it appears across the factorisations.
  4. Multiply these selected primes together to get the LCM.
💡 The LCM must contain every prime factor from every number, with the highest exponent.
Simultaneous Division Method for HCF and LCM
  1. Write the numbers side by side, separated by commas.
  2. Divide all numbers by a common prime factor (or any common factor).
  3. Write the quotients below in a new row.
  4. Repeat until the numbers in the last row have no common factor other than 1.
  5. HCF = product of all the divisors on the left side.
  6. LCM = product of all the divisors on the left side and the numbers in the last row.
💡 You can divide by any common factor, not just primes — this can make the process faster.
Exam Traps — Don't Make These Mistakes
⚠ Confusing HCF and LCM in word problems
✗ Students often use LCM when the problem asks for HCF, or vice versa.
→ HCF is used when you need the largest size that divides quantities exactly (e.g., largest tile size). LCM is used when you need the smallest common multiple (e.g., shortest toran length).
✓ Identify the key phrase: 'largest' or 'greatest' suggests HCF; 'smallest' or 'least' suggests LCM.
⚠ Forgetting 1 as a common factor
✗ Students list common factors as 2, 4, 6, etc., and omit 1.
→ 1 is always a factor of every number, so it is always a common factor.
✓ Always include 1 in the list of common factors.
⚠ Taking maximum instead of minimum for HCF
✗ For HCF of 30 and 72, students write 2×2×2×3×3 = 72 (taking maximum occurrences of 2 and 3).
→ HCF requires the minimum number of occurrences of each common prime, not the maximum. 30 has only one 2 and one 3, so HCF = 2×3 = 6.
✓ For each common prime, use the smallest exponent found in any of the numbers.
⚠ Taking minimum instead of maximum for LCM
✗ For LCM of 96 and 360, students write 2×2×2×3×5 = 120 (taking minimum occurrences).
→ LCM requires the maximum number of occurrences of each prime. 96 has five 2s, 360 has two 3s, so LCM = 2⁵×3²×5 = 1440.
✓ For each prime, use the largest exponent found in any of the numbers.
⚠ Applying HCF × LCM = Product to three numbers
✗ Students write HCF(a,b,c) × LCM(a,b,c) = a × b × c.
→ This property holds only for two numbers. For three numbers, it is not generally true.
✓ Use the property only for pairs of numbers. For three numbers, find HCF and LCM separately.
⚠ Misidentifying the 'largest common factor' in division method
✗ In the simultaneous division method, students stop when the numbers in the last row are 1 and something else, and take only the divisors as HCF.
→ The HCF is the product of all divisors on the left. The last row numbers are not part of the HCF.
✓ Multiply all the divisors (the numbers on the left side) to get the HCF.
Compare & Contrast
Definition
Highest Common Factor (HCF)
Lowest Common Multiple (LCM)
HCF is the greatest number that divides all given numbers exactly; LCM is the smallest number that is a multiple of all given numbers.
Selection of prime occurrences
HCF using prime factorisation
LCM using prime factorisation
For HCF, take the minimum number of occurrences of each common prime. For LCM, take the maximum number of occurrences of each prime that appears in any number.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 SA (3 marks), 1 LA (5 marks)
Most asked: Finding HCF and LCM using prime factorisation · Word problems involving HCF (tiling, packing) and LCM (torans, free gajak, meeting cycles) · Relationship HCF × LCM = Product of two numbers · Properties of HCF and LCM for consecutive numbers, even numbers, co-prime numbers
Don't skip: Word problems requiring application of HCF or LCM in real-life contexts (e.g., tiling, packing, torans, free gajak); Finding HCF and LCM of three numbers using prime factorisation; Verifying the relationship HCF × LCM = Product of two numbers
One-Line Revision (read aloud before the exam)
  1. HCF is the highest common factor of two or more numbers.
  2. LCM is the lowest common multiple of two or more numbers.
  3. HCF is also called GCD (Greatest Common Divisor).
  4. To find HCF, list common factors and pick the largest.
  5. To find LCM, list common multiples and pick the smallest.
  6. Prime factorisation: write a number as a product of primes.
  7. For HCF using primes: take common primes with minimum occurrences.
  8. For LCM using primes: take all primes with maximum occurrences.
  9. HCF × LCM = Product of the two numbers (for two numbers only).
  10. If one number is a factor of another, the HCF is the smaller number.
  11. If two numbers are co-prime, their HCF is 1.
  12. The LCM of two co-prime numbers is their product.
  13. In the simultaneous division method, HCF = product of divisors, LCM = product of divisors and last row numbers.
  14. A conjecture is a statement made without proof; a counterexample disproves it.
  15. A generalisation is a statement that holds true in all cases.
  16. 1 is always a common factor of any set of numbers.
  17. The largest common factor of two numbers is always less than or equal to the smaller number.
  18. The smallest common multiple of two numbers is always greater than or equal to the larger number.

MCQs — 10 Questions with Answers

Q1. What is the HCF of 20 and 30?

  • A. 5
  • B. 10 ✓
  • C. 15
  • D. 20

Answer: B — Factors of 20 are 1, 2, 4, 5, 10, 20 and factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30; the highest common factor is 10.

Q2. Which of the following is the prime factorisation of 36?

  • A. 2 × 18
  • B. 6 × 6
  • C. 2² × 3² ✓
  • D. 3² × 4

Answer: C — 36 = 2 × 2 × 3 × 3 = 2² × 3²; the other options contain composite numbers, not only primes.

Q3. What is the HCF of 84 and 108?

  • A. 6
  • B. 12 ✓
  • C. 18
  • D. 36

Answer: B — 84 = 2² × 3 × 7 and 108 = 2² × 3³; common primes are 2² and 3, so HCF = 4 × 3 = 12.

Q4. Is 15 a common factor of 45 and 75?

  • A. Yes, because 45 ÷ 15 = 3 and 75 ÷ 15 = 5 ✓
  • B. No, because 15 is too large
  • C. Yes, but only if 15 is prime
  • D. No, because 75 is larger than 45

Answer: A — 15 divides both 45 and 75 exactly, so it is a common factor; it is also the HCF.

Q5. Sameeksha's room is 12 ft by 16 ft. What is the largest square tile size (whole number) she can use?

  • A. 2 ft
  • B. 3 ft
  • C. 4 ft ✓
  • D. 6 ft

Answer: C — The tile size must divide both 12 and 16 exactly; the HCF of 12 and 16 is 4 ft.

Q6. Lekhana has 84 kg and 108 kg of rice. What is the maximum weight of each equal bag?

  • A. 6 kg
  • B. 12 kg ✓
  • C. 18 kg
  • D. 24 kg

Answer: B — The bag weight must divide both 84 and 108 equally; the HCF is 12 kg, using fewest bags.

Q7. A market vendor has 30 apples and 45 oranges. She wants to pack them into identical fruit baskets with the same number of each fruit in each basket. What is the maximum number of baskets she can make?

  • A. 5
  • B. 10
  • C. 15 ✓
  • D. 30

Answer: C — The HCF of 30 and 45 is 15, so she can make 15 baskets with 2 apples and 3 oranges each.

Q8. Two ropes are 56 m and 84 m long. They need to be cut into equal-length pieces with no waste. What is the longest possible length of each piece?

  • A. 14 m
  • B. 28 m ✓
  • C. 42 m
  • D. 56 m

Answer: B — 56 = 2³ × 7 and 84 = 2² × 3 × 7; common primes are 2² and 7, so HCF = 4 × 7 = 28 m.

Q9. Which pair of numbers has HCF equal to 1?

  • A. 20 and 30
  • B. 24 and 36
  • C. 15 and 28 ✓
  • D. 18 and 27

Answer: C — 15 = 3 × 5 and 28 = 2² × 7 share no common prime factors, so their HCF is 1.

Q10. A school has 60 pencils and 90 erasers. These must be packed into identical gift boxes with equal numbers of pencils and erasers in each box. What is the maximum number of gift boxes that can be made?

  • A. 15
  • B. 20
  • C. 30 ✓
  • D. 45

Answer: C — HCF of 60 and 90 is 30; so 30 boxes can be made with 2 pencils and 3 erasers each.

Flashcards

What is the HCF of 12 and 16?

The HCF of 12 and 16 is 4, because 4 is the largest number that divides both 12 and 16 exactly.

Write the prime factorisation of 90.

The prime factorisation of 90 is 2 × 3 × 3 × 5.

What are the common factors of 45 and 75?

The common factors of 45 and 75 are 1, 3, 5, and 15.

Define a prime number.

A prime number is a number greater than 1 that has only 1 and itself as its factors.

What is the HCF of 96 and 275?

The HCF of 96 and 275 is 1, because they share no common prime factors.

How do you find HCF using prime factorisation?

Find common primes in both numbers, then multiply the minimum count of each common prime together.

What is a counterexample in mathematics?

A counterexample is a specific example that proves a conjecture false.

What is the HCF of 30 and 72 using their prime factorisations?

Since 30 = 2 × 3 × 5 and 72 = 2³ × 3², the HCF is 2 × 3 = 6.

Are the prime factors of a number always the same?

Yes, the prime factorisation of any number is unique, though the order of primes may change.

In Sameeksha's room problem, why is 4 ft the best tile size?

Because 4 ft is the HCF of 12 ft and 16 ft, making it the largest size that fits both dimensions exactly with fewest tiles.

Important Board Questions

What is the HCF of 12 and 16? [1 mark]

List factors of both numbers or use prime factorisation: 12 = 2² × 3 and 16 = 2⁴; common prime is 2², so HCF = 4.

Find all common factors of 45 and 75. [2 marks]

Prime factorise both: 45 = 3² × 5 and 75 = 3 × 5²; common subparts are 1, 3, 5, and 15. List them in order.

Sameeksha wants to tile her floor measuring 12 ft by 16 ft with square tiles of the largest possible size. How many tiles will she need? Show your working. [3 marks]

Step 1: Find HCF of 12 and 16 (= 4 ft). Step 2: Calculate number of tiles along length = 16 ÷ 4 = 4 tiles. Step 3: Calculate number of tiles along breadth = 12 ÷ 4 = 3 tiles. Step 4: Total = 4 × 3 = 12 tiles.

A shopkeeper has 84 kg of rice from one farm and 108 kg from another. She wants to pack them in equal-weight bags (whole kg) such that each bag contains rice from only one farm. (a) What should be the weight of each bag to use the fewest bags? (b) How many bags of each type will she need? Show all steps. [5 marks]

Step 1: Find HCF of 84 and 108 using prime factorisation: 84 = 2² × 3 × 7 and 108 = 2² × 3³; HCF = 2² × 3 = 12 kg. Step 2: Number of bags from farm 1 = 84 ÷ 12 = 7 bags. Step 3: Number of bags from farm 2 = 108 ÷ 12 = 9 bags. Step 4: Total bags = 7 + 9 = 16 bags. State both answers clearly.

Assertion-Reason Questions for Finding Common Ground

Q1. Assertion (A):

The HCF of 12 and 16 is 4.

Reason (R):

4 is the largest number that divides both 12 and 16 exactly.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. The HCF is defined as the highest common factor, and 4 divides both 12 and 16 evenly (12÷4=3, 16÷4=4), and no number larger than 4 divides both. The reason correctly explains why 4 is the HCF.

Q2. Assertion (A):

Prime factorisation of a number is unique regardless of the order in which we factor.

Reason (R):

Prime numbers can be multiplied in different orders to give the same result.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — Both statements are true. The prime factorisation is unique (Fundamental Theorem of Arithmetic)—a number has one unique set of prime factors. However, the reason explains commutativity of multiplication rather than uniqueness of prime factorisation. The reason is not the correct explanation for why factorisation is unique; it's just about reordering factors, which is a property of multiplication, not a proof of uniqueness.

Q3. Assertion (A):

To find all factors of 225 = 5² × 3², we can systematically form combinations of the prime factors.

Reason (R):

Any product made from a subset of the prime factors of a number is also a factor of that number.

Answer: A — Both A and R are true, and R is the correct explanation of A — The assertion is true—we get factors like 1, 3, 5, 9, 15, 25, 45, 75, 225 by combining prime factors. The reason is true and correctly explains the assertion. Every subset product of prime factors divides the original number exactly, so it is a factor. This is the correct mathematical principle behind finding all factors via prime factorisation.

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Case-Based Questions (CBQ) for Finding Common Ground

Case Study:

A community centre is organising a sports carnival with 72 cricket balls and 60 badminton shuttlecocks. The centre director wants to prepare identical prize kits containing both items in equal quantities from the available stock. Each kit must have a whole number of each item, and no items should be left unpacked. The director wants to make the maximum number of kits possible so more students can win prizes. Help determine the maximum number of kits that can be prepared and the contents of each kit.

Q1. What is the maximum number of prize kits that can be prepared?

  • A. 12 kits ✓
  • B. 6 kits
  • C. 36 kits
  • D. 60 kits

Answer: A

Q2. How many cricket balls will be in each kit?

  • A. 5 cricket balls
  • B. 6 cricket balls ✓
  • C. 8 cricket balls
  • D. 12 cricket balls

Answer: B

Q3. How many badminton shuttlecocks will be in each kit?

  • A. 4 shuttlecocks
  • B. 5 shuttlecocks ✓
  • C. 6 shuttlecocks
  • D. 10 shuttlecocks

Answer: B

Q4. Which mathematical concept is used to find the maximum number of kits?

  • A. Highest Common Factor (HCF) ✓
  • B. Least Common Multiple (LCM)
  • C. Prime factorisation only
  • D. Simple division

Answer: A

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