LCM of two numbers is 28 times their HCF. The sum of the HCF and the LCM is…

Concept

For any two positive integers, the product of the two numbers always equals the product of their LCM and their HCF: first number × second number = LCM × HCF. When the LCM is given as a fixed multiple of the HCF (here, LCM = 28 × HCF), the HCF can be represented as a single variable and the LCM derived from it.

Application

1. Let the HCF of the two numbers be x. Since LCM = 28 × HCF, the LCM is 28x.

2. It is given that HCF + LCM = 1740, so x + 28x = 1740 ⇒ 29x = 1740 ⇒ x = 60.

3. So HCF = 60 and LCM = 28 × 60 = 1680.

4. Using the product relationship, first number × second number = LCM × HCF: 240 × second number = 1680 × 60 = 100800.

5. Solving for the second number: second number = 100800 ÷ 240 = 420.

Cross-check

• 240 = 24 × 3 × 5 and 420 = 22 × 3 × 5 × 7.

• HCF(240, 420) = 22 × 3 × 5 = 60 and LCM(240, 420) = 24 × 3 × 5 × 7 = 1680, matching the values found above.

• HCF + LCM = 60 + 1680 = 1740, which matches the given condition.

Hence, the other number is 420.