If the sum of 2 numbers is 185, their LCM is 1700 and HCF is 5. Then the…

Concept

For any two positive integers, the product of their HCF and LCM equals the product of the two numbers: HCF × LCM = (first number) × (second number). Also, if the HCF is h, each number is a multiple of h, so the numbers can be written as h·a and h·b where a and b are co-prime.

Application

1. Let the numbers be 5a and 5b (HCF = 5, with a and b co-prime).

2. Sum: 5a + 5b = 185, so a + b = 37.

3. Product rule: HCF × LCM = product, so 5 × 1700 = (5a)(5b) = 25ab, giving ab = 340.

4. With a + b = 37 and ab = 340, a and b are the roots of x2 − 37x + 340 = 0.

5. Discriminant = 372 − 4·340 = 1369 − 1360 = 9, so the square root is 3 and x = (37 ± 3)/2 = 20 or 17.

6. Hence the numbers are 5·20 = 100 and 5·17 = 85, and their difference is 100 − 85 = 15.

Cross-check

Check the pair 100 and 85: sum = 185 (correct), HCF = 5 (correct), and LCM = 1700 (correct), since 100 = 22·52 and 85 = 5·17, so LCM = 22·52·17 = 1700. The difference is 15.