Finding Common Ground — Class 7 Mathematics

This chapter is about the Highest Common Factor (HCF) and the Lowest Common Multiple (LCM), built on prime factorisation. A prime has only 1 and itself as factors; every number can be written as a product of primes in essentially one way (e.g. ), found efficiently by the division method. From a prime factorisation you can list all factors as products of "subparts". The HCF is the largest common factor — taken by including the minimum number of times each shared prime occurs across the numbers — while the LCM is the smallest common multiple — taken by including the maximum number of occurrences of each prime. You learn a quick column method that produces both at once, and explore general patterns (the HCF of two consecutive numbers is 1; if one number is a factor of another, it is their HCF and the larger is their LCM). A beautiful relation ties them together: for any two numbers, equals the product of the two numbers. The chapter also introduces the ideas of conjecture, counterexample and generalisation.