In this numerical analogy question, select the related number from the given alternatives to complete the proportion: 39 : 52 :: 51 : ?

Introduction / Context:
This numerical analogy question uses multiplication with a fixed factor to build a consistent pattern. The pair 39 and 52 is provided, and you must find the number that corresponds to 51 in the same way. Unlike simple addition or subtraction, this problem involves recognising that both numbers in each pair are multiples of a common prime, which makes it a slightly more challenging reasoning exercise.

Given Data / Assumptions:
The first pair is 39 and 52.
The second pair starts with 51, and we must find the related number.
The options are 53, 40, 68, 38, and 65.
We assume that there is a common structure behind 39 and 52 and that the same structure must be applied to 51 and the unknown number.

Concept / Approach:
First, factorise the numbers in the given pair. The number 39 can be written as 3 * 13, and 52 can be written as 4 * 13. This suggests that the underlying pattern is to take a common prime factor, here 13, and multiply it by 3 and then by 4. To extend this pattern, we look at the second starting number 51. Factorising 51 gives 3 * 17. To maintain the same pattern, we would expect the partner of 51 to be 4 * 17, since 39 corresponds to 3 * 13 and 52 to 4 * 13. This structure leads directly to the correct answer.

Step-by-Step Solution:
Step 1: Factorise the numbers in the first pair. 39 = 3 * 13, and 52 = 4 * 13.Step 2: Observe that both numbers share the same prime factor 13, and the multipliers are 3 and 4.Step 3: Recognise the pattern: first number is 3 times a prime, second number is 4 times the same prime.Step 4: Factorise 51. We have 51 = 3 * 17.Step 5: To keep the same structure, we should multiply 17 by 4 to obtain the partner of 51.Step 6: Compute 4 * 17 = 68.Step 7: Check the options and confirm that 68 is listed, so it is the only number that fits the pattern.

Verification / Alternative check:
We can also check the analogy using ratios. In the first pair, 52 / 39 = (4 * 13) / (3 * 13) = 4 / 3. Thus, the multiplier from the first to the second number is 4/3. Applying the same multiplier to 51, we get 51 * (4 / 3) = 17 * 4 = 68. This confirms again that 68 preserves the same proportional relationship between the terms, so it is the correct answer.

Why Other Options Are Wrong:
Option 53 is a prime number that does not relate to 51 through a factor of 4/3 or a shared prime factor structure similar to 13.
Option 40 cannot be expressed as 4 times 17 or in a form that mirrors the 3 times prime and 4 times prime pattern used in the first pair.
Option 38 lies close to 39 but does not match any ratio such as 4/3 when related to 51, so it fails the analogy.
Option 65 equals 5 * 13 and does not maintain the pairing of 3 times prime and 4 times the same prime that we observed for 39 and 52 with factor 13 and for 51 and 68 with factor 17.

Common Pitfalls:
Many candidates glance at the numbers and attempt to use simple addition or subtraction, for example noticing that 39 increases by 13 to become 52 and then incorrectly adding 13 to 51. While this sometimes works in easier questions, here it does not lead to a consistent factor structure involving a common prime and the 4/3 ratio. The more robust strategy is to factorise the numbers to see if a hidden multiplicative pattern exists, especially when two numbers share a meaningful common factor.

Final Answer:
Therefore, in the analogy 39 : 52 :: 51 : ?, the number that correctly completes the pair is 68.