The difference between two numbers is 14.\nTheir L.C.M. is 441 and their H.C.F. is 7.\nWhat are the two numbers?

Introduction:
This question tests multiple number theory concepts at once: the relationship between H.C.F., L.C.M., and the numbers themselves, along with the given difference between the two numbers. Solving it requires both algebraic reasoning and understanding of prime factorisation.

Given Data / Assumptions:

• Difference between the two numbers = 14.
• L.C.M. of the numbers = 441.
• H.C.F. of the numbers = 7.
• We need to find the two numbers.

Concept / Approach:
Let the two numbers be a and b. Since their H.C.F. is 7, we can write: a = 7x and b = 7y where gcd(x, y) = 1. Also: LCM(a, b) = 7xy = 441 and a - b = 14 ⇒ 7|x - y| = 14 ⇒ |x - y| = 2. We must find co-prime integers x and y that satisfy these conditions.

Step-by-Step Solution:
Step 1: From LCM, 7xy = 441. Step 2: So xy = 441 / 7 = 63. Step 3: From the difference, |a - b| = 14 ⇒ |7x - 7y| = 14. Step 4: Therefore, 7|x - y| = 14 ⇒ |x - y| = 2. Step 5: We need co-prime positive integers x and y with product 63 and difference 2. Step 6: Factor pairs of 63: (1,63), (3,21), (7,9). Step 7: Only the pair (7,9) has difference 2 and gcd(7,9) = 1. Step 8: Therefore, x = 7 and y = 9 (or vice versa). Step 9: The numbers are a = 7x = 49 and b = 7y = 63. Step 10: So the two numbers are 49 and 63.

Verification / Alternative check:
Check the conditions: Difference: 63 - 49 = 14 (matches). H.C.F.(49, 63) = 7. LCM(49, 63) = (49 * 63) / 7 = 441. All given conditions are satisfied, so the answer is correct.

Why Other Options Are Wrong:
64 and 48: Difference is 16, and H.C.F. is not 7. 62 and 46: Difference 16 again, and their H.C.F. is not 7. 64 and 49: Difference 15, not 14. 56 and 42: Difference 14, but H.C.F. is 14 and L.C.M. is not 441.

Common Pitfalls:
Some learners forget to factor out the H.C.F. into the form 7x and 7y, or do not enforce that x and y must be co-prime. Others may not systematically check factor pairs of 63 for the required difference and coprimality, leading to incorrect guesses.

Final Answer:
The two numbers are 63 and 49.