6
CBSE Marks
★★★★★
Difficulty
8
Topics
High
Board Weight
Topics Covered
8 key topics in this chapter
Euclid's Division Lemma
Euclid's Division Algorithm
Fundamental Theorem of Arithmetic
HCF & LCM via Prime Factorisation
Revisiting Irrational Numbers
Proof: √2, √3 are Irrational
Decimal Expansions of Rationals
Terminating & Non-Terminating Decimals
Study Resources
Key Formulas
| Formula / Rule | Expression |
|---|---|
| Euclid's Division | \(a = bq + r, 0 ≤ r < b\) |
| HCF via Factorisation | \(\text{HCF = product of smallest powers of common primes}\) |
| LCM via Factorisation | \(\text{LCM = product of greatest powers of all prime factors}\) |
| HCF–LCM Relation | \(\mathrm{HCF(a,b) × LCM(a,b) = a × b}\) |
Important Points to Remember
HCF × LCM = Product of two numbers (for any two positive integers).
Every composite number can be expressed as a product of primes in a unique way — this is the Fundamental Theorem of Arithmetic.
A rational number p/q (in lowest terms) has a terminating decimal expansion if and only if q has no prime factor other than 2 or 5.
√2, √3, √5 and all surds of the form √p (p prime) are irrational — proved by contradiction.