The sum and difference of the LCM and HCF of two numbers are 592 and 518 respectively. If the sum of the numbers be 296, then the numbers are :

Question

Accepted Answer

To solve the problem, we need to find two numbers whose sum is 296, and the sum and difference of their LCM (Least Common Multiple) and HCF (Highest Common Factor) are 592 and 518, respectively. Let's break down the solution step by step:

1. **Understanding the Given Information:**
   - Let the two numbers be $ a $ and $ b $.
   - We know that $ a + b = 296 $.
   - Let the LCM of $ a $ and $ b $ be $ L $ and the HCF be $ H $.
   - According to the problem, $ L + H = 592 $ and $ L - H = 518 $.

2. **Solving for LCM and HCF:**
   - We have two equations:
     $$
     L + H = 592
     $$
     $$
     L - H = 518
     $$
   - Adding these two equations, we get:
     $$
     2L = 1110 \implies L = 555
     $$
   - Subtracting the second equation from the first, we get:
     $$
     2H = 74 \implies H = 37
     $$

3. **Using the Relationship Between LCM, HCF, and the Numbers:**
   - The relationship between the LCM, HCF, and the numbers is given by:
     $$
     L \times H = a \times b
     $$
   - Substituting the values we found:
     $$
     555 \times 37 = a \times b
     $$
     $$
     a \times b = 20535
     $$

4. **Finding the Numbers:**
   - We have two equations now:
     $$
     a + b = 296
     $$
     $$
     a \times b = 20535
     $$
   - These are the equations of a quadratic:
     $$
     x^2 - 296x + 20535 = 0
     $$
   - Solving this quadratic equation using the quadratic formula:
     $$
     x = \frac{296 \pm \sqrt{296^2 - 4 \times 20535}}{2}
     $$
     $$
     x = \frac{296 \pm \sqrt{87616 - 82140}}{2}
     $$
     $$
     x = \frac{296 \pm \sqrt{5476}}{2}
     $$
     $$
     x = \frac{296 \pm 74}{2}
     $$
   - This gives us two solutions:
     $$
     x = \frac{370}{2} = 185 \quad \text{and} \quad x = \frac{222}{2} = 111
     $$

5. **Conclusion:**
   - The numbers are 111 and 185, which matches Option 3.

6. **Verification of Other Options:**
   - **Option 1 (107 and 111):** The sum is 218, not 296.
   - **Option 2 (111 and 158):** The sum is 269, not 296.
   - **Option 4 (158 and 167):** The sum is 325, not 296.

Therefore, the correct answer is indeed Option 3: 111 and 185.