In this number analogy question, 13 is related to 17, so you must select the pair of numbers that stands in the same relationship from the given alternatives.

Introduction / Context:

This is a number analogy that tests basic knowledge of prime numbers and their patterns. The pair "13 : 17" connects two special numbers that both happen to be prime and separated by a fixed difference. Your task is to find a similar pair among the options that maintains the same type of relationship.

Given Data / Assumptions:

- First pair: 13 and 17.
- Candidate pairs: 17 : 24, 19 : 23, 19 : 27, 21 : 23 and 15 : 21.
- Prime numbers are integers greater than 1 that have no positive divisors other than 1 and themselves.
- We assume that the analogy is based on simple properties like primality and numerical difference.

Concept / Approach:

We first look at what is special about 13 and 17. Both are prime numbers, and their difference is 17 minus 13, which equals 4. Thus, 13 and 17 form a pair of prime numbers with a difference of 4. We now scan the option pairs to find another pair where both numbers are prime and the difference between them is also 4.

Step-by-Step Solution:

Verification / Alternative check:

We can list nearby prime numbers to confirm the pattern: 11, 13, 17, 19, 23 and 29. Looking at these, we see that some consecutive primes differ by 2, some by 4 or more. The pair 13 and 17 is a prime pair with a difference of 4, and similarly, 19 and 23 form the next such pair in this sequence. No other option reproduces the same combination of properties. This confirms that 19 : 23 is the correct analogy.

Why Other Options Are Wrong:

17 : 24 fails because 24 is not a prime number and the difference is 7, not 4.

19 : 27 fails because 27 is composite and the difference is 8, so the pair does not mirror the original pattern.

21 : 23 has one composite and one prime number with a smaller difference of 2, so it does not match the structure of the first pair.

15 : 21 consists of two composite numbers and therefore does not follow the prime-number theme at all.

Common Pitfalls:

A common error is to focus only on the difference and ignore whether the numbers are prime. For example, one might be tempted by any pair with a difference of 4 even if one of the numbers is composite. It is important to identify all key properties of the first pair, not just one. Another pitfall is misidentifying prime numbers, so candidates should be comfortable checking divisibility for numbers up to at least 30.

Final Answer:

The pair that mirrors the relationship of prime numbers differing by four is 19 : 23.