Find two natural numbers whose sum is 85 and whose least common multiple (LCM) is 102.

Introduction / Context:
This problem tests understanding of the relationship between the greatest common divisor, least common multiple and the product of two numbers. It also requires checking candidate pairs against conditions on both sum and least common multiple. Such questions are standard in number system topics of aptitude exams.

Given Data / Assumptions:
- We are looking for two natural numbers. - Their sum must be 85. - Their least common multiple is 102. - Several candidate pairs are suggested in the options.

Concept / Approach:
- For two positive integers a and b, the relation a * b = gcd(a, b) * lcm(a, b) always holds. - However, here it may be faster to test the given options directly. - For each candidate pair, check if the sum is 85 and then compute its least common multiple. - The pair that simultaneously satisfies both conditions is the correct answer.

Step-by-Step Solution:
Step 1: Check option A, 30 and 55. The sum is 30 + 55 = 85, so the sum condition is satisfied. Step 2: Compute the greatest common divisor of 30 and 55. Their common factors are 1 and 5, so gcd(30, 55) = 5. Step 3: Use the relation lcm(a, b) = (a * b) / gcd(a, b) = (30 * 55) / 5 = 330. This does not match the required 102. Step 4: Check option B, 17 and 68. Sum is 17 + 68 = 85, sum condition holds. Step 5: Compute gcd(17, 68). Since 17 is a prime factor of 68, gcd(17, 68) = 17. Then lcm is (17 * 68) / 17 = 68, not 102. Step 6: Check option C, 35 and 55. Sum is 90, which already fails the sum condition 85, so this pair can be rejected immediately. Step 7: Check option D, 51 and 34. Sum is 51 + 34 = 85, so the sum condition is satisfied. Step 8: Compute gcd(51, 34). Both are divisible by 17, so gcd(51, 34) = 17. Step 9: Compute lcm(51, 34) using the product relation: lcm = (51 * 34) / 17 = (1734) / 17 = 102. Step 10: Since this pair satisfies both the sum and LCM conditions, 51 and 34 are the required numbers.

Verification / Alternative check:
We can also factor 102 as 2 * 3 * 17. The pair 51 and 34 are 3 * 17 and 2 * 17 respectively. Their product is 51 * 34 = 1734. Dividing by their gcd 17 gives 102, and their sum is 85, confirming that they perfectly meet both conditions.

Why Other Options Are Wrong:
Option A (30 and 55): Although the sum is 85, their LCM is 330, not 102. Option B (17 and 68): Sum equals 85 but the LCM is only 68. Option C (35 and 55): The sum is 90, which breaks the sum requirement. Option E (49 and 36): The sum is 85 but gcd and LCM do not yield 102. The LCM would be larger than 102 in this case.

Common Pitfalls:
- Assuming that matching the sum alone is enough and ignoring the LCM condition. - Miscalculating the greatest common divisor and hence getting the wrong LCM. - Forgetting the useful identity a * b = gcd(a, b) * lcm(a, b) which simplifies such problems.

Final Answer:
The required natural numbers are 51 and 34.