The L.C.M. of two numbers is 495 and their H.C.F. is 5. If the sum of the two numbers is 100, what is the difference between these two numbers?

Introduction / Context:
This question combines three important number theory ideas: sum of two numbers, their highest common factor (H.C.F.), and their least common multiple (L.C.M.). Such problems test your understanding of the relationship between product, H.C.F., and L.C.M., as well as your ability to solve simple quadratic equations arising from these relationships.

Given Data / Assumptions:

• L.C.M. of the two numbers = 495.
• H.C.F. of the two numbers = 5.
• Sum of the two numbers = 100.
• We must find the difference between the two numbers.

Concept / Approach:
For any two positive integers a and b, the product a * b is equal to H.C.F. * L.C.M. when H.C.F. and L.C.M. are known. Using this, we can find the product of the numbers. Together with the sum, we can set up a quadratic equation whose roots are the two numbers. Once the numbers are found, subtract the smaller from the larger to get the required difference.

Step-by-Step Solution:
Step 1: Let the two numbers be x and y.Step 2: Given H.C.F. = 5 and L.C.M. = 495.Step 3: Use the relation x * y = H.C.F. * L.C.M. = 5 * 495 = 2475.Step 4: Also given x + y = 100.Step 5: Let x and y be roots of the quadratic t^2 - (sum)t + product = 0.Step 6: So t^2 - 100t + 2475 = 0.Step 7: Compute the discriminant: D = 100^2 - 4 * 2475 = 10000 - 9900 = 100.Step 8: Roots are t = (100 ± √100) / 2 = (100 ± 10) / 2.Step 9: Thus, t = 55 or t = 45, so the two numbers are 55 and 45.Step 10: The difference between the two numbers is 55 - 45 = 10.

Verification / Alternative check:
Check H.C.F. of 55 and 45: gcd(55, 45) = 5.Check L.C.M.: (55 * 45) / 5 = 2475 / 5 = 495.Check sum: 55 + 45 = 100. All conditions are satisfied, confirming that the solution is consistent and the difference is 10.

Why Other Options Are Wrong:
Option b (46), option c (70), and option d (90) correspond to wrong pairs of numbers that do not satisfy all three conditions simultaneously. If you try to reconstruct possible pairs with those differences, you will see that they either fail the product condition or the H.C.F. and L.C.M. relation.

Common Pitfalls:
Candidates sometimes forget the key formula relating product, H.C.F., and L.C.M., or they miscompute the product. Another common mistake is solving the quadratic incorrectly or mixing up the roots. Always double-check the discriminant and verify that both the sum and product match the original data for the found numbers.

Final Answer:
The difference between the two numbers is 10.