Class 7 Maths (Part II) Chapter 3 : Finding Common Ground | HCF, LCM, Prime Factorization & Real-Life Problems 2025
Complete Chapter 3 (Part II) guide: finding Highest Common Factor (HCF) for tiling floors and packing rice bags, finding Lowest Common Multiple (LCM) for cloth torans and 'Jump Jackpot' games, using prime factorization (division method and factor trees) to find HCF and LCM easily, solving word problems involving scheduling (sweet shop visit) and resource optimization, plus solved examples and practice questions for CBSE Class 7 Maths
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Class 7 Mathematics Chapter 3: Finding Common Ground – HCF and LCM Complete Notes, Solutions, Questions & Answers 2025
Finding Common Ground
Class 7 Mathematics Chapter 3 | Complete Guide | HCF and LCM 2025
Chapter at a Glance – Finding Common Ground
This chapter explores Highest Common Factor (HCF) and Lowest Common Multiple (LCM) using real-life examples, prime factorization, and properties.
Main Topics Covered
HCF using tile and rice bag examples
Prime numbers and prime factorization
Finding factors using prime factorization
HCF using prime factorization
LCM using toran and gajak examples
LCM using prime factorization
Patterns and properties of HCF and LCM
Efficient procedures for HCF and LCM
Relationship between HCF and LCM
Conjectures, generalizations, and counterexamples
Key Takeaways for Exams
HCF Definition
Highest common factor of numbers.
LCM Definition
Lowest common multiple of numbers.
Prime Factorization
Break down into primes.
HCF Method
Min exponents of common primes.
LCM Method
Max exponents of all primes.
Property
HCF × LCM = Product of numbers.
Division Method
For prime factorization.
Efficient Procedure
Divide by common factors for HCF/LCM.
Key Rules & Properties – HCF and LCM
Important methods and properties for calculating HCF and LCM.
HCF Using Prime Factorization
Take the minimum power of each common prime factor.
(a) Consecutive even: 2 (b) Consecutive odd: 1 (c) Even: At least 2 (d) Consecutive: 1 (e) Co-prime: 1
2. LCM one of numbers
When one is factor of other.
3. General statements for LCM
(a) Multiples of 3: Multiple of 3 (b) Consecutive even: 2 × odd LCM (c) Consecutive: Product if co-prime (d) Co-prime: Product
Final Figure it Out
1. Blue stars meet
LCM of periods.
2(a) Multiple?
No
2(b) Factor?
No
3(a) HCF and LCM
HCF = 3×7×7=147, LCM = 3²×5×7²×11×12
3(b) 45 and 36
HCF = 9, LCM = 180
4. Numbers HCF 1 LCM 66
6 and 11
5. Cows <200, equal through 3,5,7 gates
LCM of 3,5,7 = 105
6. Cubes in box 12x18x36
(b) 6 cm, (d) 3 cm, (e) 2 cm
7. Largest divides 306 and 36
(c) 18
8. Smallest divisible by 3,4,5,7 remainder 10 /11
LCM 3,4,5,7=420, find k*420 +10 ≡0 mod 11, etc.
9. Fire in Mountain, multiples
(a) 72
10. LCM primes m,n
(c) Greater than both, (d) Less than m×n
11. Dog rabbit leaps
Relative speed 9-7=2, 150/2=75
12. Smallest multiple 1-6,8-10
LCM=2520
13. Sum fractions
LCM denominators, add.
Extra Practice Questions (Exam-Ready) – Chapter 3
25+ Questions • Categorized by Marks • With Detailed Solutions • Difficulty Tags
1-Mark Questions (Very Short Answer)
1. HCF of 12 and 18.
6
2. LCM of 4 and 6.
12
3. Prime factors of 30.
2 × 3 × 5
4. HCF of co-primes.
1
5. LCM × HCF = ?
Product
2-Mark Questions (Short Answer)
6. Factors of 225.
1, 3, 5, 9, 15, 25, 45, 75, 225
7. HCF of 50 and 60.
10
8. LCM of 30 and 72.
360
9. Consecutive numbers HCF.
1
10. Prime factorization 1200.
2^4 × 3 × 5^2
3-Mark Questions (Reasoning)
11. Explain HCF using primes for 45,75.
Min powers: 3^1 × 5^1 =15
12. Why HCF * LCM = product.
From prime factors distribution.
13. Jump size for 14,30.
HCF=2
14. LCM for co-primes.
Product
15. Counterexample for longer factorization.
96 longer than 121.
4–5 Mark Questions (Application)
16. Tile size for 12x16 room.
4 ft
17. Bag weight for 84,108 kg rice.
12 kg
18. Toran length for 6,8 cm strips.
24 cm
19. Gajak next in 70 days.
LCM 7,10=70
20. HCF 240,378.
6
Challenge Questions (6+ Marks)
21. Prove HCF*LCM=product.
Using prime factors.
22. Numbers HCF1 LCM66.
6,11
23. Cows <200, divisible 3,5,7.
105
24. Smallest multiple 1-10 except7.
2520
25. Dog rabbit 9:7, 150 ft.
75 leaps
Common Mistakes & How to Avoid
Mistake 1: Listing All Factors for Large Numbers
Missing some factors leading to wrong HCF.
Avoid: Use prime factorization method.
Mistake 2: Confusing Min/Max in HCF/LCM
Using max for HCF or min for LCM.
Avoid: Remember min for HCF, max for LCM.
Mistake 3: Forgetting to Include All Primes in LCM
Missing unique primes.
Avoid: List all primes from both factorizations.
Mistake 4: Wrong Assumption for Consecutive Numbers
Thinking HCF >1.
Avoid: Consecutive are co-prime, HCF=1.
Mistake 5: Not Verifying HCF*LCM=Product
Calculation error undetected.
Avoid: Always check with property.
Mistake 6: Incomplete Division in Efficient Method
Stopping too early.
Avoid: Continue until no common factors.
History & Fun Facts
Ancient Origins
The Euclidean algorithm for finding the GCD (HCF) was described by Euclid around 300 BC in his work "Elements." It is one of the oldest algorithms still in use today.
The concept of LCM has roots in ancient mathematics, with contributions from Babylonian and Greek mathematicians for solving problems involving fractions and multiples.
Real-Life Applications
Scheduling: LCM for repeating events like bus timings.
Resource Allocation: HCF for dividing items equally.
Cryptography: Primes and factors in encryption.
Music: LCM for rhythm synchronization.
Fun Facts
HCF is always a factor of LCM.
The product of HCF and LCM of two numbers equals their product.
Euclid's algorithm can find HCF without factorization for large numbers.
In computing, HCF/LCM used in fraction reduction, scheduling algorithms.
Largest known prime has over 24 million digits, impacts factorization.
Did You Know?
Brahmagupta (7th century) contributed to arithmetic with positives and negatives, related to factors.
