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Take and . Which observation explains why HCF(a, b) × LCM(a, b) always equals ?

Class 10Real NumbersNumber Systems
  • ABecause both the HCF and the LCM are multiples of a
  • BBecause for each prime, the LOWER exponent (used by the HCF) and the HIGHER exponent (used by the LCM) are exactly the two original exponents, so they add up to their sum
  • CBecause a and b are co-prime
  • DIt is a coincidence that happens to work for these particular numbers

Step-by-Step Solution

Here HCF = and LCM = ·5. For the prime 2 the exponents 2 and 3 are just the original pair reused; the same holds for 3 and for 5 (exponents 0 and 1). Multiplying HCF by LCM therefore rebuilds exactly. These two numbers are NOT co-prime, so that option is false as well as irrelevant.
Answer: B) Because for each prime, the LOWER exponent (used by the HCF) and the HIGHER exponent (used by the LCM) are exactly the two original exponents, so they add up to their sum

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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