Chapter 11 — Finding Common Ground

The chapter 'Finding Common Ground' covers the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of numbers. Students learn how to find them efficiently using prime factorisation, discover the relationship HCF × LCM = product of the two numbers, and explore related patterns and properties.

The Highest Common Factor (HCF) of two or more numbers is the highest (or greatest) of their common factors. It is also known as the Greatest Common Divisor (GCD).

The Lowest Common Multiple (LCM) of two or more given numbers is the lowest (or smallest or least) of their common multiples.

Write the prime factorisation of each number using the division method. Identify the primes common to all factorisations and take the minimum number of times each common prime appears across any of the factorisations. The product of these gives the HCF. For example, 30 = 2 × 3 × 5 and 72 = 2 × 2 × 2 × 3 × 3 share primes 2 and 3; the minimum occurrences are one 2 and one 3, so HCF = 2 × 3 = 6.

Write the prime factorisation of each number. Collect all prime factors that appear in any of the numbers and, for each prime, take the maximum number of times it appears in any single factorisation. The product of these gives the LCM. For example, LCM of 96 (= 2⁵ × 3) and 360 (= 2³ × 3² × 5) is 2⁵ × 3² × 5 = 1440.

Prime factorisation means writing a number as a product of primes. In the division method, you repeatedly divide the number by a prime factor, writing the quotients below, and continue until you reach a prime. The prime factors are the divisors used and the final prime at the bottom. For example, 105 = 3 × 5 × 7.

Yes. Regardless of how you start factorising a number, the resulting prime factors are always the same — perhaps in a different order, but identical in set and count. For example, 90 started as 3 × 30 or 2 × 45 always ends at 2 × 3 × 3 × 5.

For two numbers, HCF × LCM = Product of the two numbers. For example, the LCM of 105 and 95 multiplied by their HCF (which is 5) equals 105 × 95.

The HCF also doubles. When both numbers are doubled, each gets an extra factor of 2 in its prime factorisation, which is included in the largest common subpart. For example, HCF of 270 and 50 is 10, while HCF of 540 and 100 (their doubles) is 20.

A conjecture is a statement or claim made without proof or verification. The chapter gives the example of Anshu's claim that 'the larger a number is, the longer its prime factorisation will be,' which is disproved by a counterexample: 96 = 2 × 2 × 2 × 2 × 2 × 3 (six factors) is smaller than 121 = 11 × 11 (two factors).

Yes. Divide both numbers simultaneously by a common prime factor and write the quotients in the next row, continuing until no common prime factor remains. The product of all the divisors used is the HCF; the LCM is the product of all divisors and the two final quotients. Dividing by composite common factors (not just primes) makes this even faster.

The HCF of two numbers is one of the numbers when that number is a factor of the other — equivalently, when one number is a multiple of the other. For example, the HCF of 6 and 18 is 6 because 6 is a factor of 18.

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