If the sum of 2 numbers is 165, their LCM is 1300 and HCF is 5. Then the…

Concept

For any two positive numbers, the product of the two numbers equals the product of their LCM and HCF. The difference of two numbers can then be recovered from their sum and product, because the difference squared equals the sum squared minus four times the product.

(difference)2 = (sum)2 − 4 × (product), and product = LCM × HCF.

Application

1. Find the product of the two numbers: product = LCM × HCF = 1300 × 5 = 6500.

2. Apply the identity: (difference)2 = (sum)2 − 4 × product = 1652 − 4 × 6500.

3. Evaluate each term: 1652 = 27225 and 4 × 6500 = 26000, so (difference)2 = 27225 − 26000 = 1225.

4. Take the square root: difference = √1225 = 35.

Cross-check

Solving for the numbers themselves gives 100 and 65: their sum is 100 + 65 = 165, their product is 6500, their HCF is 5 and their LCM is 1300. All three conditions hold, and 100 − 65 = 35 confirms the difference.