Consider the pair series (2, 3), (3, 5), (5, 7), (7, 11), (11, 13), ___. What is the next ordered pair of numbers that should follow this pattern?

Introduction / Context:
This question gives a sequence of ordered pairs and asks for the next pair. It is a pattern-recognition problem based on prime numbers. Such questions appear in reasoning and aptitude tests to check familiarity with basic number properties such as primes and the ability to notice how they are arranged in a given pattern.

Given Data / Assumptions:

• The given pairs are (2, 3), (3, 5), (5, 7), (7, 11), (11, 13).
• We must find the next pair in the same format, (a, b).
• All entries are positive integers.
• The pattern is consistent and is related to prime numbers.

Concept / Approach:
We observe that each pair seems to consist of two prime numbers. Moreover, from one pair to the next, the second element of the previous pair often becomes the first element of the next pair. This creates a kind of overlapping sequence of consecutive primes. So we list consecutive prime numbers and see how they form the pairs.

Step-by-Step Solution:
Step 1: List the first few prime numbers in order: 2, 3, 5, 7, 11, 13, 17, 19, and so on.Step 2: Note the given pairs: (2, 3), (3, 5), (5, 7), (7, 11), (11, 13).Step 3: In each pair, both numbers are primes. Also, except for the first pair, the first number of each pair is the second number of the previous pair.Step 4: Observe the sequence of primes in the first and second positions: 2, 3, 5, 7, 11, 13.Step 5: Continuing with the prime sequence after 13, the next prime is 17.Step 6: Following the pattern, the next pair should start with 13 (the second element of the last given pair) and then use the next prime, which is 17.Step 7: Therefore, the next pair is (13, 17).

Verification / Alternative check:
Check that the pattern is consistent: pair 1 uses primes 2 and 3, pair 2 uses primes 3 and 5, pair 3 uses primes 5 and 7, pair 4 uses primes 7 and 11, and pair 5 uses primes 11 and 13. The first element of each new pair is the second element of the previous pair, and the second element is the next prime in the sequence. Following this rule naturally produces (13, 17) as the next pair.

Why Other Options Are Wrong:
Option 13, 15: The number 15 is not a prime number, so it breaks the pattern of using only primes.
Option 15, 17: Again, 15 is not prime, so this pair does not follow the established structure.
Option 13, 19: This skips the prime 17, so it does not use the next prime after 13.
Option 17, 19: This does not continue from the second element of the previous pair, which should be 13, so it breaks the link between consecutive pairs.

Common Pitfalls:
Some learners may focus only on differences between the numbers, such as 1, 2, 2, 4, 2, and not recognise the prime number pattern. Others might think the second number always increases by 2, which is only true among some primes but not when the gap becomes larger, as between 7 and 11. Recognising primes and their natural order is crucial here.

Final Answer:
The next ordered pair in the series is 13, 17.