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.
Primes and prime factorisationListing factors from prime factorsFinding the HCFFinding the LCMHCF–LCM relation, patterns and generalisation
Key terms
Prime number
A number greater than 1 whose only factors are 1 and itself.
Prime factorisation
Writing a number as a product of primes, e.g. \(90 = 2\times3\times3\times5\).
Highest Common Factor (HCF)
The greatest factor common to two or more numbers; also the GCD.
Lowest Common Multiple (LCM)
The smallest multiple common to two or more numbers.
Co-prime numbers
Two numbers whose only common factor is 1, so their HCF is 1.
Conjecture
A statement believed true but not yet proved; disproved by a counterexample.
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The HCF of two numbers is:
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