Real Numbers (HCF/LCM) Questions

Prime factors, HCF and LCM, and why √2 is irrational

Real numbers include rational numbers, which can be written as p/q with q ≠ 0, and irrational numbers such as √2, √3 and π, whose decimal expansions never end or repeat. The fundamental theorem of arithmetic states that every composite number can be written as a product of primes in exactly one way, apart from the order of the factors; for example, 360 = 2³ × 3² × 5. Euclid's division algorithm finds the HCF of two numbers by repeated division: divide, keep the remainder, and repeat until the remainder is zero.

Using prime factorisation, the HCF is the product of the smallest powers of the common primes, and the LCM is the product of the greatest powers of all the primes involved. For any two positive integers, HCF × LCM equals the product of the numbers, so knowing one helps you find the other. To prove that √2 is irrational, assume √2 = p/q in lowest terms; squaring gives p² = 2q², which forces p and then q to be even, contradicting the lowest-terms assumption.

Why it matters

HCF and LCM solve practical problems such as cutting ribbons into the longest equal pieces, finding when two bells will next ring together, or tiling a floor with the largest possible square tiles. Prime factorisation also underlies the encryption that protects online banking and private messages. Proof by contradiction, met here for the first time, is a method used throughout higher mathematics.

Key terms

Fundamental theorem of arithmetic

Every composite number has a unique factorisation into primes, apart from order.

HCF

The highest common factor, the largest number that divides each of the given numbers exactly.

LCM

The lowest common multiple, the smallest number that is a multiple of each given number.

Irrational number

A real number that cannot be written as a fraction of two integers.