Real Numbers
Euclid's division, HCF/LCM, and why √2 refuses to be a fraction.
Every integer breaks down into primes in exactly one way — the Fundamental Theorem of Arithmetic. HCF × LCM = product of the two numbers. And √2 is irrational because assuming it's p/q leads to a contradiction: both p and q would end up even, forever.
Real numbers is about how whole numbers are built from primes, and why some numbers can never be written as neat fractions.
It gives you two tools — Euclid's Division Lemma and the Fundamental Theorem of Arithmetic — that quietly power HCF, LCM, and irrationality proofs.
You use these ideas whenever you simplify a fraction, split things into equal groups, or check if a decimal terminates.
Euclid's Division Lemma
For positive integers a and b, a = bq + r where 0 ≤ r < b. Repeating this with remainders shrinking each time gives Euclid's algorithm for HCF — it always ends because remainders keep getting smaller.
Fundamental Theorem of Arithmetic
Every integer greater than 1 is a product of primes, and this factorisation is unique except for the order of the primes. This is why HCF and LCM can be found by comparing prime powers.
HCF, LCM and their link
HCF takes the lowest power of each common prime; LCM takes the highest power of every prime present. For any two numbers a and b: HCF(a,b) × LCM(a,b) = a × b.
Irrational numbers
A number is irrational if it cannot be written as p/q with q ≠ 0. Proofs like '√2 is irrational' assume the opposite and show it forces both p and q to share a common factor forever — a contradiction.
These are easy on purpose — they check understanding, not speed. Nothing here is scored.
1. What does the Fundamental Theorem of Arithmetic guarantee?
2. Why does Euclid's division algorithm always stop?
3. A number that can be written as p/q (q≠0) is called:
HCF of 96 and 404 is…