In the number pairs 14, 12; 24, 7; 42, 4 and 37, 4, find the odd pair based on whether the second number is prime or composite.

Introduction / Context:
This odd one out question involves number pairs where the key lies in recognizing whether the second number in each pair is prime or composite. Rather than complex operations linking the two numbers, the pattern is based on a simple property of the second number. Such questions appear in aptitude tests to evaluate knowledge of basic number theory ideas along with pattern recognition skills.

Given Data / Assumptions:
The number pairs are 14, 12; 24, 7; 42, 4 and 37, 4. We focus on the nature of the second number in each pair. A prime number is a natural number greater than 1 that has exactly two distinct positive divisors, 1 and itself. A composite number has more than two positive divisors.

Concept / Approach:
The key concept is to classify the second number of each pair as prime or composite. If three of the second numbers are composite and one is prime, then the pair with the prime second number is the odd one out. This approach is much simpler and more reliable than attempting to relate the first and second numbers through complex formulas. Checking primality for small integers is quick and ideal for exam conditions.

Step-by-Step Solution:
Step 1: Look at the pair 14, 12. The second number is 12. Its factors include 1, 2, 3, 4, 6 and 12, which means it has more than two divisors and is a composite number. Step 2: Look at the pair 24, 7. The second number is 7. Its only positive divisors are 1 and 7, so it is a prime number. Step 3: Look at the pair 42, 4. The second number is 4. Its positive divisors are 1, 2 and 4, so it has more than two divisors and is composite. Step 4: Look at the pair 37, 4. Again the second number is 4, which we already identified as a composite number. Step 5: Summarize the classification of the second numbers: 12 is composite, 7 is prime, 4 is composite and 4 is composite. Step 6: Three pairs have a composite second number, while one pair has a prime second number. The pair with the prime second number is the odd one out. Step 7: Therefore, the pair 24, 7 is the odd pair in this set.

Verification / Alternative check:
As a verification, list the second numbers only: 12, 7, 4 and 4, then mark which are prime. Only 7 qualifies as a prime number, because it has no divisors other than 1 and 7. The numbers 12 and 4 both have multiple divisors and are clearly composite. Since the odd one out must be unique, the pair containing 7 as the second number is clearly intended as the different one.

Why Other Options Are Wrong:
14, 12 is not the odd pair because the second number 12 is composite, just like 4 in the other composite pairs. 42, 4 is not the odd pair because the second number 4 is composite. 37, 4 is not the odd pair because its second number is again the composite number 4. 24, 7 is the odd pair because its second number 7 is the only prime among the second numbers.

Common Pitfalls:
A frequent mistake is to try to relate each first number to its second using operations such as subtraction, division or ratios, which can be time consuming and may not reveal any clear rule. Another pitfall is to misidentify 1 or small numbers as primes without checking the definition carefully. To avoid errors, recall that primes greater than 5 usually end in 1, 3, 7 or 9, and always verify the number of divisors when you are unsure.

Final Answer:
The pair whose second number is prime whereas the others have composite second numbers, and which is therefore the odd one out, is 24, 7.