The difference of two positive integers is 14. Their LCM is 441 and their HCF (GCD) is 7. Find the two numbers.

Introduction / Context:This problem combines several important facts: the relationship between two numbers and their HCF and LCM, the idea of factoring out the HCF, and the constraint on the difference between the two numbers. It is a classic type of HCF and LCM reasoning question.

Given Data / Assumptions:

• Difference between the numbers = 14
• HCF = 7
• LCM = 441
• Numbers are positive integers

Concept / Approach:Let the two numbers be 7a and 7b because 7 is the HCF. Then gcd(a, b) = 1, which means a and b are coprime. The LCM of 7a and 7b is 7ab. We are told:LCM = 7ab = 441 ⇒ ab = 63Also, the difference is 14:|7a - 7b| = 14 ⇒ |a - b| = 2We must find a and b such that a * b = 63, |a - b| = 2, and gcd(a, b) = 1.

Step-by-Step Solution:Step 1: From 7ab = 441, divide by 7: ab = 63.Step 2: From 7|a - b| = 14, divide by 7: |a - b| = 2.Step 3: List factor pairs of 63: (1, 63), (3, 21), (7, 9).Step 4: Check each pair for difference 2 and gcd = 1.(1, 63): difference 62, so not suitable.(3, 21): difference 18 and gcd(3, 21) = 3, so not suitable.(7, 9): difference 2 and gcd(7, 9) = 1, so this pair works.Step 5: Thus a = 7, b = 9 (or vice versa). The numbers are 7a = 49 and 7b = 63.

Verification / Alternative check:Check HCF and LCM: gcd(49, 63) = 7, and LCM(49, 63) = (49 * 63) / 7 = 441. The difference is |63 - 49| = 14. All conditions match the given data.

Why Other Options Are Wrong:48 and 64: HCF is 16, and LCM is not 441.46 and 62: HCF is 2, not 7.56 and 70: Difference is 14, but HCF is 14 and LCM is 280.42 and 56: HCF is 14, again not equal to 7, and the LCM does not match 441.

Common Pitfalls:Not factoring out the HCF first, which makes it harder to use the LCM relation.Ignoring the requirement that a and b must be coprime after factoring out the HCF.Selecting factor pairs of 63 that do not satisfy the difference constraint of 2.

Final Answer:49 and 63