NCERT Solutions for Class 7th Maths Chapter 3 Finding Common Ground
Updated on 2026-09-19
About this chapter
The Highest Common Factor (HCF) of two or more numbers is the greatest of their common factors. It is also called the Greatest Common Divisor (GCD). The Lowest Common Multiple (LCM) is the least of their common multiples. There is no greatest common multiple — the list of common multiples never ends. Every factor of a number is a subpart of its prime factorisation. So a common factor must be a subpart of both factorisations, and the HCF is the largest such subpart. HCF rule: keep only the primes common to all the numbers, each taken the minimum number of times it occurs. LCM rule: keep every prime that occurs anywhere, each taken the maximum number of times. The ladder (division) method divides both numbers by a common factor again and again. The left column multiplies to the HCF; the left
- The Greatest of All
- Primes and Prime Factorisation
- Factors of a Number Using Prime Factorisation
- Finding the HCF of Numbers Using Prime Factorisation
- Finding the HCF Directly from the Prime Factorisations
- Least, but not Last!
- Finding LCM through Prime Factorisation
- Patterns, Properties, and a Pretty Procedure!
- Doubling both numbers · Multiples of the same number
- Efficient Procedures for HCF and LCM
Quick revision
| Idea | What it means | Example from the chapter | Result |
|---|---|---|---|
| Common factor | A number that divides both numbers | factors of 12 and 16 | 1, 2, 4 |
| HCF (GCD) | The greatest common factor | 12 and 16 (Sameeksha's room) | 4 ft tile |
| Prime factorisation | Rewrite composite factors till only primes remain | 90 = 2 × 3 × 3 × 5 | unique, up to order |
| Division method | Divide by a prime, bring the quotient down, repeat | 105 → 3, 5, 7 | 105 = 3 × 5 × 7 |
| Factor = subpart | Any group of the prime factors multiplies to a factor | 840 = (2 × 2 × 7) × (2 × 3 × 5) | 28 is a factor |
| HCF by primes | Common primes, each taken the minimum number of times | 30 = 2 × 3 × 5; 72 = 2 × 2 × 2 × 3 × 3 | HCF = 6 |
| LCM by primes | All primes, each taken the maximum number of times | 96 = 2⁵ × 3; 360 = 2³ × 3² × 5 | LCM = 1440 |
| Ladder method | Divide both by a common factor, repeat | 630, 770 → 2, 5, 7 → 9, 11 | HCF 70, LCM 6930 |
| One number is a factor of the other | Then it is the HCF and the other is the LCM | n and 5n | HCF n, LCM 5n |
| Co-prime numbers | No common prime factor | 7 and 11 | HCF 1, LCM 77 |
| Product property | HCF × LCM = the product of the two numbers | 105 × 95 = 5 × (3 × 5 × 7 × 19) | HCF 5, LCM 1995 |
| Conjecture | A claim made without proof; one counterexample kills it | “Larger number → longer prime factorisation” | 121 = 11 × 11 |
Exercises
- In-text Questions — The Greatest of All Page 47
- In-text Questions — The Greatest of All Page 48
- Try This — The Greatest of All Page 48
- In-text Questions — Primes and Prime Factorisation Page 49
- In-text Questions — The Division Method · Factors of a Number Using Prime Factorisation Page 50
- In-text Questions — Factors of a Number Using Prime Factorisation Page 51
- Figure it Out — Factors of a Number Using Prime Factorisation Page 51
- In-text Questions — Finding the HCF of Numbers Using Prime Factorisation Page 51–53
- Figure it Out — Finding the HCF of Numbers Using Prime Factorisation Page 53
- In-text Questions — Finding the HCF Directly from the Prime Factorisations Page 53–54
- Figure it Out — Finding the HCF of Numbers Using Prime Factorisation Page 54
- In-text Questions — Least, but not Last! Page 55
- In-text Questions — Least, but not Last! Page 56
- In-text Questions — Finding LCM through Prime Factorisation Page 57–58
- Figure it Out — Finding LCM through Prime Factorisation Page 58
- In-text Questions — Patterns, Properties, and a Pretty Procedure! Page 58–59
- Figure it Out — Patterns, Properties, and a Pretty Procedure! Page 59
- In-text Questions — Doubling both numbers · Multiples of the same number Page 59–60
- In-text Questions — Efficient Procedures for HCF and LCM Page 60–61
- Try This — Efficient Procedures Page 62
- In-text Questions — Property Involving both the HCF and the LCM Page 62–63
- Try This — Property Involving both the HCF and the LCM Page 63
- End-of-chapter question set — Figure it Out Page 63–64
- Summary — In-text Question Page 65
- Puzzle Time