If the sum of 2 numbers is 175, their LCM is 300 and HCF is 25. Then the…

Concept

For any two positive integers, HCF × LCM = product of the two numbers. Also, if H is their HCF, both numbers are multiples of H, so they can be written as Hx and Hy where x and y are co-prime (their HCF is 1).

Application

1. Write the numbers as 25x and 25y, since the HCF is 25 and x, y are co-prime.

2. Use HCF × LCM = product: 25 × 300 = (25x)(25y), so 7500 = 625xy, giving xy = 12.

3. Use the sum: 25x + 25y = 175, so x + y = 7.

4. Find co-prime integers with x + y = 7 and xy = 12: these are 3 and 4, whose HCF is 1.

5. Recover the numbers: 25 × 3 = 75 and 25 × 4 = 100.

6. Difference = 100 − 75 = 25.

Cross-check

Verify 75 and 100 against all three conditions: sum = 75 + 100 = 175, HCF(75, 100) = 25, LCM(75, 100) = 300. All match, and the pair is unique, so the difference is 25.