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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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.
| Numbers | Prime Factors | HCF |
|---|---|---|
| 30, 72 | 30 = 2 × 3 × 5, 72 = 2³ × 3² | 2 × 3 = 6 |
| 225, 750 | 225 = 3² × 5², 750 = 2 × 3 × 5³ | 3 × 5² = 75 |
LCM Using Prime Factorization
Take the maximum power of each prime factor.
| Numbers | Prime Factors | LCM |
|---|---|---|
| 14, 35 | 14 = 2 × 7, 35 = 5 × 7 | 2 × 5 × 7 = 70 |
| 96, 360 | 96 = 2⁵ × 3, 360 = 2³ × 3² × 5 | 2⁵ × 3² × 5 = 1440 |
Properties
- HCF × LCM: Equals product of two numbers.
- For multiples: HCF(a, ka) = a if k integer.
- Consecutive numbers: HCF = 1.
- Co-primes: HCF = 1, LCM = product.
- Doubled numbers: HCF doubles if both doubled.
Concept Cards – Quick Explanations
HCF
Highest common factor, largest common divisor.
LCM
Lowest common multiple, smallest common multiple.
Prime Numbers
Numbers >1 with factors 1 and itself.
Prime Factorization
Express as product of primes.
Division Method
For prime factorization.
Factors from Primes
All combinations of prime powers.
Conjecture
Unproven statement, disprove with counterexample.
Generalization
Statement holding for all cases.
Efficient HCF/LCM
Divide by common factors step by step.
HCF * LCM = Product
Key relation for two numbers.
Co-prime Numbers
HCF = 1.
Mystery Colors
Color scheme based on factors/primes.
Examples + Solutions
Example 1: HCF of 45 and 75
Prime Factors: 45 = 3 × 3 × 5, 75 = 3 × 5 × 5
Solution: Common: 3, 5. HCF = 3 × 5 = 15
Example 2: HCF of 112 and 84
Prime Factors: 112 = 2⁴ × 7, 84 = 2² × 3 × 7
Solution: HCF = 2² × 7 = 28
Example 3: HCF of 96 and 275
Prime Factors: 96 = 2⁵ × 3, 275 = 5² × 11
Solution: No common primes, HCF = 1
Example 4: HCF of 30 and 72
Prime Factors: 30 = 2 × 3 × 5, 72 = 2³ × 3²
Solution: HCF = 2 × 3 = 6
Example 5: HCF of 225 and 750
Prime Factors: 225 = 3² × 5², 750 = 2 × 3 × 5³
Solution: HCF = 3 × 5² = 75
Example 6: LCM of 14 and 35
Prime Factors: 14 = 2 × 7, 35 = 5 × 7
Solution: LCM = 2 × 5 × 7 = 70
Example 7: LCM of 96 and 360
Prime Factors: 96 = 2⁵ × 3, 360 = 2³ × 3² × 5
Solution: LCM = 2⁵ × 3² × 5 = 1440
Figure it Out Solutions (All Solved)
Factors of Numbers
(a) 90
1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
(b) 105
1, 3, 5, 7, 15, 21, 35, 105
(c) 132
1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132
(d) 360 (24 factors)
1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360
(e) 840 (32 factors)
1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, 840
Common Factors and HCF
(a) 50, 60
Common: 1, 2, 5, 10. HCF = 10
(b) 140, 275
Common: 1, 5. HCF = 5
(c) 77, 725
Common: 1. HCF = 1
(d) 370, 592
Common: 1, 2. HCF = 2
(e) 81, 243
Common: 1, 3, 9, 27, 81. HCF = 81
More HCF
(a) 24, 180
HCF = 12
(b) 42, 75, 24
HCF = 3
(c) 240, 378
HCF = 6
(d) 400, 2500
HCF = 100
(e) 300, 800
HCF = 100
LCM
(a) 30, 72
LCM = 360
(b) 36, 54
LCM = 108
(c) 105, 195, 65
LCM = 1365
(d) 222, 370
LCM = 1110
Figure it Out 3.3
1. General statements for HCF
(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.
2. LCM of 4 and 6.
3. Prime factors of 30.
4. HCF of co-primes.
5. LCM × HCF = ?
2-Mark Questions (Short Answer)
6. Factors of 225.
7. HCF of 50 and 60.
8. LCM of 30 and 72.
9. Consecutive numbers HCF.
10. Prime factorization 1200.
3-Mark Questions (Reasoning)
11. Explain HCF using primes for 45,75.
12. Why HCF * LCM = product.
13. Jump size for 14,30.
14. LCM for co-primes.
15. Counterexample for longer factorization.
4–5 Mark Questions (Application)
16. Tile size for 12x16 room.
17. Bag weight for 84,108 kg rice.
18. Toran length for 6,8 cm strips.
19. Gajak next in 70 days.
20. HCF 240,378.
Challenge Questions (6+ Marks)
21. Prove HCF*LCM=product.
22. Numbers HCF1 LCM66.
23. Cows <200, divisible 3,5,7.
24. Smallest multiple 1-10 except7.
25. Dog rabbit 9:7, 150 ft.
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.
Quick Revision One-Pager
HCF and LCM Methods
| Method | HCF | LCM |
|---|---|---|
| Prime Fact | Min powers common primes | Max powers all primes |
| Efficient | Product of common divisors | Product of all in grid |
Quick Rules
- ✓ HCF: Largest common divisor
- ✓ LCM: Smallest common multiple
- ✓ Prime Fact: Division method
- ✓ Property: HCF × LCM = a × b
- ✓ Co-prime: HCF=1, LCM=product
- ✓ Consecutive: HCF=1
Mind Map
Central: HCF & LCM
- Basics: Factors, multiples
- Methods:
- Prime factorization
- Efficient division
- Properties: Patterns, generalizations
- Applications: Real-life problems
Exam Tips
Before Solving
Prime factorize first
During Solving
Check min/max powers
After Solving
Verify with property
Time-Savers
Use efficient method
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