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

Introduction:
This problem tests your understanding of the relationship between the highest common factor (H.C.F.), least common multiple (L.C.M.), and the product of two numbers. It also involves forming and solving a quadratic equation using the sum and product of the numbers, a common pattern in aptitude exams.

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 are required to find the difference between the two numbers.

Concept / Approach:
For any two positive integers a and b, we use the identity: a * b = HCF(a, b) * LCM(a, b) Once the product and sum of the numbers are known, we can take a and b as roots of a quadratic equation: x^2 - (sum)x + (product) = 0 Solving this equation gives the two numbers, and their difference can then be computed easily.

Step-by-Step Solution:
Step 1: Let the two numbers be a and b. Step 2: Product a * b = HCF * LCM = 5 * 495 = 2475. Step 3: Sum a + b = 100 (given). Step 4: Form the quadratic: x^2 - (a + b)x + a * b = 0. Step 5: So, x^2 - 100x + 2475 = 0. Step 6: Compute the discriminant: D = 100^2 - 4 * 2475 = 10000 - 9900 = 100. Step 7: Roots are x = (100 ± 10) / 2 → x = 55 and x = 45. Step 8: The difference between the numbers = 55 - 45 = 10.

Verification / Alternative check:
Check the conditions: Sum = 55 + 45 = 100, product = 55 * 45 = 2475. Also, H.C.F.(55,45) = 5 and L.C.M.(55,45) = 2475 / 5 = 495. All conditions are satisfied, so the solution is consistent.

Why Other Options Are Wrong:
90: This would require the numbers to be 5 and 95, which do not have L.C.M. 495. 70: Would correspond to numbers like 15 and 85, again not matching the given L.C.M. and H.C.F. 46: No pair of integers satisfying the product and L.C.M. conditions gives this difference. 55: This would imply one of the numbers is zero or negative in this context, which is not valid.

Common Pitfalls:
A common mistake is to forget the product relation with L.C.M. and H.C.F., or to incorrectly compute the quadratic discriminant. Some students also try trial and error with random pairs instead of using a systematic algebraic approach.

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