CBSE Class 7 Maths Chapter 3 Finding Common Ground Notes 2025-26

Finding common ground in mathematics is about discovering patterns and connections between numbers through concepts like factors, multiples, HCF (Highest Common Factor), and LCM (Lowest Common Multiple). This is essential not only for problem solving but also for understanding how numbers fit together in real-world situations. The chapter begins with practical scenarios—such as tiling a room or dividing rice—demonstrating how HCF and LCM help in daily life.

Greatest Common Factors: Real-World Examples

To select the largest size for square tiles in a room that measures 12 ft by 16 ft, the side length should be a common factor of both numbers. The common factors of 12 and 16 are 1, 2, and 4. Using the greatest one (4 ft) means fewer, larger tiles and no cutting or gaps. The principle is the same for dividing items like packing rice into equal-weight bags—choose the highest weight that evenly divides both quantities.

• The HCF of two numbers is the largest number that divides both without leaving a remainder.
• Common factors can be listed, but for large numbers, prime factorisation is easier and more accurate.
• Other real-life uses include optimising packaging, event planning cycles, and matching patterns.

Prime factorisation means breaking a number down into a product of prime numbers. For example, 90 becomes 2 × 3 × 3 × 5. This technique simplifies identifying all factors of a number and is a reliable method to find both HCF and LCM, especially for bigger numbers where manually listing factors becomes complicated. When writing any number as a product of its primes, the result is unique except for the order of the factors.

• A prime number has only two factors: 1 and itself.
• Prime factorisation can be done using the division method—dividing repeatedly by the smallest prime possible.
• All the factors of a number can be found by taking products of its prime factors in every possible way.

To find the HCF using prime factors, write each number as a product of its primes and choose only the common prime factors, taking the smallest power for each. For instance, for 30 (2 × 3 × 5) and 72 (2 × 2 × 2 × 3 × 3), the HCF is 2 × 3 = 6. If two numbers have no common primes, their HCF is 1, making them coprime.

Some examples from the chapter:

• HCF of 225 and 750 is 75.
• For 112 and 84, HCF is 28.
• When numbers share no common factor but 1, such as 96 and 275, HCF is 1.

LCM is the smallest number that is a common multiple of two or more numbers and is widely used for finding repetition cycles (like events, schedules, or packing). It’s easy to see with context, for example, when two people with steps of 6 and 8 will “meet” at the same place—they meet at every 24th step (their LCM).

• Find LCM by listing common multiples, or use prime factorisation: include every prime that appears, using the highest power from any number.
• LCM of 14 and 35 is 70. For 96 and 360, it's 1,440.

Several patterns are seen in HCF and LCM:

• If one number is a factor of another, their HCF is the smaller number, and their LCM is the bigger number.
• The product of HCF and LCM of two numbers is always equal to the product of those two numbers.
• If both numbers are doubled, so is their HCF.
• If two numbers are coprime, their HCF is 1, and their LCM is just their product.

Besides prime factorisation, you can find HCF by dividing both numbers repeatedly by common factors starting from the smallest (like 2, then 3, etc.), multiplying the divisors used. The same method extends to more complex numbers. This makes calculation faster and reduces errors when working with several numbers at once.

• For example, finding HCF of 84 and 180 by successive division gives 2, 2, and 3 as common factors, and the HCF is their product, 12.

The chapter introduces conjectures (claims not proven), generalisations, and practical questions to strengthen understanding.

• A conjecture is an educated guess in mathematics not yet proven.
• Generalization means applying a pattern or property to all similar cases.
• “Coprime” numbers have HCF of only 1. “Multiple” means a number can be produced by multiplying another by an integer.

The chapter is full of “Try This” and “Figure It Out” questions asking students to list factors, compute HCF or LCM, or solve real-life context scenarios such as, “Which sized cube will fill a box of specific dimensions exactly?” or “Find the smallest number divisible by a set of numbers but having a fixed remainder by another.” These encourage real active revision and practical understanding.

There are also fun pattern activities (like colouring schemes or number games) to help develop a deeper insight into properties of numbers and build math intuition.

• Last year’s topics—common multiples, common factors, primes, and factorisation—are revised here with real examples.
• HCF is the maximum of common factors; LCM is the minimum of common multiples.
• For HCF, use the lowest power of shared primes; for LCM, use the highest power of all appearing primes.
• Fast methods for calculation help avoid errors when numbers grow larger.
• Important: HCF × LCM = Product of the two numbers.
• Learn the difference between conjectures and proved properties.

These revision notes give not just formulas but also engaging examples, games, and challenges to grow your problem-solving skills and confidence in dealing with factors, multiples, primes, HCF, and LCM.

Class 7 Maths Chapter 3 Finding Common Ground Notes – Easy Revision for HCF and LCM

Mastering Class 7 Maths Chapter 3: Finding Common Ground is much easier with these structured revision notes. All key topics like HCF, LCM, and prime factorisation are covered clearly for fast learning and better understanding. These notes empower you to solve everyday math problems confidently.

From practical examples to step-by-step methods, these Class 7 Maths Finding Common Ground notes will help you revise all important concepts quickly. Grasping patterns and properties in numbers is now simple and fun, preparing you for any exam question on HCF, LCM, and related problems.