Chapter 3: Finding Common GroundClass 7 Mathematics — summary, notes, extra questions & MCQ quiz

Summary

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. \(90 = 2\times3\times3\times5\)), 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, \(\text{HCF}\times\text{LCM}\) equals the product of the two numbers. The chapter also introduces the ideas of conjecture, counterexample and generalisation.

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