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Does the rule HCF × LCM = product of the numbers extend to THREE numbers, that is, is HCF(a, b, c) × LCM(a, b, c) = a × b × c?

Class 10Real NumbersNumber Systems
  • AYes, it holds for any number of integers
  • BYes, provided the three numbers are all even
  • CNo — take 2, 3 and 4: HCF = 1 and LCM = 12, but while
  • DNo, because the HCF of any three numbers is always 1

Step-by-Step Solution

The identity is proved only for TWO numbers, and a single counterexample settles the question: HCF(2, 3, 4) = 1, LCM(2, 3, 4) = 12, so the left side is 12 but the product is 24. The last option is a false reason — HCF(4, 6, 8) = 2, not 1.
Answer: C) No — take 2, 3 and 4: HCF = 1 and LCM = 12, but while

Key idea: Fundamental Theorem of Arithmetic

**Theorem**: Every composite number can be expressed as a product of primes — UNIQUELY, up to the order of factors.

Example: . There is no other set of primes whose product is 360.

**Consequences**:
1. **Finding HCF** by primes: HCF = product of LOWEST powers of common primes.
2. **Finding LCM** by primes: LCM = product of HIGHEST powers of all primes appearing.
3. **Identity**: HCF(a,b) × LCM(a,b) = .

Example: . HCF = 2·3 = 6, LCM = . Check: . ✓

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