If the ratio of two numbers is 2 : 9 and their HCF is 22, then their LCM is:

Concept

When two numbers are in the ratio a : b (in lowest terms) with HCF h, the numbers are a × h and b × h. Since a and b share no common factor, and for any two numbers, HCF × LCM equals the product of the numbers, the LCM equals a × b × h.

Application

1. The ratio 2 : 9 is already in lowest terms, since 2 and 9 share no common factor other than 1.

2. With HCF = 22, the two numbers are 2 × 22 = 44 and 9 × 22 = 198.

3. Product of the two numbers = 44 × 198 = 8712.

4. Using HCF × LCM = product of the numbers: LCM = 8712 ÷ 22 = 396.

Cross-check

Cross-check by prime factorization:

Reading off the common and highest powers: HCF = 2 × 11 = 22, which matches the given HCF, and LCM = 22 × 32 × 11 = 396, which matches the computed result.

Result

Hence the LCM of the two numbers is 396.