Final answer:
To find the two numbers, we can set up a system of equations using the given information. Using the fact that the LCM multiplied by the HCF is equal to the product of the two numbers, we can solve for the HCF and find the values of the numbers.
Step-by-step explanation:
The given information states that the sum of the least common multiple (LCM) and highest common factor (HCF) of two numbers is 592, while their difference is 518. Also, the sum of the two numbers is 296. Let's denote the two numbers as a and b.
From the information provided, we can set up the following equations:
a + b = 296 (equation 1)
LCM(a, b) + HCF(a, b) = 592 (equation 2)
We can use the fact that LCM(a, b) * HCF(a, b) = a * b to solve this system of equations. Rearranging equation 2, we get:
LCM(a, b) = 592 - HCF(a, b) (equation 3)
Substituting equation 3 into the LCM * HCF equation:
(592 - HCF(a, b)) * HCF(a, b) = a * b
Expanding and rearranging:
592HCF(a, b) - (HCF(a, b))^2 = ab
Since we know the sum of the two numbers is 296 (a + b = 296), we can rearrange equation 1 to express a in terms of b:
a = 296 - b
Substituting this into the previous equation:
592HCF(a, b) - (HCF(a, b))^2 = (296 - b)b
This is a quadratic equation in terms of HCF(a, b). Solving this equation will give us the value of HCF(a, b), and we can find the values of a and b using equation 1.