Real Numbers

Find HCF and LCM of 12 and 18 using prime factorization method. Verify that HCF × LCM = Product of two numbers. [CBSE 2025]

Prove that √5 is irrational. [CBSE 2024]

Express 140 as product of its prime factors. [CBSE 2023]

Use Euclid's division algorithm to find HCF of 867 and 255. [CBSE 2022]

Write whether 2√45 + 3√20/2√5 on simplification gives rational or irrational number. [CBSE 2021]

For any positive integers a and b, there exist unique integers q and r such that: a = bq + r, where 0 ≤ r < b

For any two positive integers a and b: HCF(a,b) × LCM(a,b) = a × b

Every composite number can be expressed as a product of primes, and this factorization is unique (apart from order)

A number that can be expressed in the form p/q where p and q are integers and q ≠ 0

Numbers that cannot be expressed in the form p/q, examples: √2, √3, √5, π, e

Assume √2 = p/q (in lowest terms). Then 2 = p²/q², so 2q² = p². This means p² is even, therefore p is even. Let p = 2m. Then 2q² = 4m², so q² = 2m². This means q is also even. But this contradicts our assumption that p/q is in lowest terms. Hence √2 is irrational.

Apply division lemma: a = bq + r. If r = 0, HCF = b. If r ≠ 0, apply lemma to b and r. Repeat until remainder becomes 0. The divisor at this stage is the HCF.

Assume √2 = p/q (rational in lowest terms). Then 2 = p²/q², so 2q² = p². This means p² is even, so p is even. Let p = 2m. Then 2q² = 4m², so q² = 2m². This means q is also even. But this contradicts our assumption that p/q is in lowest terms. Therefore √2 is irrational.

To find HCF of a and b: Apply division lemma a = bq + r. If r = 0, HCF = b. If r ≠ 0, apply division lemma to b and r. Repeat until remainder = 0.