Two numbers are in the ratio 3:4. The product of their HCF and LCM is 2028.…

Concept

For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b.

If two numbers are in the ratio m:n, write them as mk and nk, where k is their common scale factor.

Application

1. Write the numbers as 3k and 4k.

2. Use the product identity: (3k)(4k) = 2028, so 12k2 = 2028.

3. Divide by 12: k2 = 169. Since the numbers are positive, k = 13.

4. Therefore, the numbers are 3 × 13 = 39 and 4 × 13 = 52, and their sum is 39 + 52 = 91.

Cross-check

The HCF of 39 and 52 is 13 and their LCM is 156; 13 × 156 = 2028, so both the ratio and the product condition are satisfied.