Can two numbers have 18 as their HCF and 380 as their LCM? Give the reason.

Class 10Real NumbersNumber Systems

AYes, because 380 > 18
BNo, because HCF must divide LCM and 18 does not divide 380
CYes, because both are even
DNo, because LCM must be prime

Step-by-Step Solution

The HCF of two numbers must always divide their LCM. 380 ÷ 18 is not an integer

(380 = 18 \times 21 + 2)

, so such numbers cannot exist.

Answer: B) No, because HCF must divide LCM and 18 does not divide 380

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:

360 = 2^{3} \times 3^{2} \times 5

. 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) =

a \times b

.

Example:

a = 12 = 2^{2} \cdot 3, b = 18 = 2 \cdot 3^{2}

. HCF = 2·3 = 6, LCM =

2^{2} \cdot 3^{2} = 36

. Check:

6 \times 36 = 216 = 12 \times 18

. ✓

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