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Assertion (A): HCF(a, b) × LCM(a, b) = a × b holds for all positive integers a and b. Reason (R): The HCF of two numbers is always less than or equal to their LCM. Which option is correct?

Class 10Real NumbersNumber Systems
  • ABoth A and R are true, and R is the correct explanation of A
  • BBoth A and R are true, but R is NOT the correct explanation of A
  • CA is true but R is false
  • DA is false but R is true

Step-by-Step Solution

A is true (product identity). R is also true (HCF ≤ LCM). But R does not explain why the product identity holds; they are independent facts.
Answer: B) Both A and R are true, but R is NOT the correct explanation of A

Key idea: Euclid's Division Lemma and HCF

**Lemma**: For positive integers a, (with ), there exist UNIQUE with and 0 ≤ .

This is just formal long division — is the quotient, is the remainder.

**Algorithm to find HCF(a, b)** with :
1. Apply lemma: .
2. If , HCF = .
3. If ≠ 0, replace (a, b) with , r) and repeat.

**Example**: HCF(420, 130)
→ HCF = 10.

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