Each option shows an ordered pair of two digit numbers. Select the odd pair based on whether the difference between the two numbers is prime or composite: (64, 23), (42, 12), (63, 18), (83, 20).

Introduction / Context:
This odd one out question presents pairs of two digit numbers. The hidden pattern involves examining the difference between the two numbers in each pair and checking whether that difference is a prime number or a composite number. Three pairs share one property, while one pair behaves differently. Such problems test knowledge of prime and composite numbers as well as basic subtraction.

Given Data / Assumptions:

• Pairs given are (64, 23), (42, 12), (63, 18), (83, 20), and (55, 16).
• We consider the absolute difference between the numbers in each pair.
• A prime number has exactly two positive divisors, 1 and itself.
• A composite number has more than two positive divisors.
• We will classify the difference for each pair as prime or composite.

Concept / Approach:
For each pair, we compute the difference as the larger number minus the smaller number. Then we test whether that difference is prime by checking if it has any factors other than 1 and itself. If most of the differences are composite, and one is prime, the pair with the prime difference is the odd one out, or vice versa. In this question, three differences are composite numbers, and only one difference is prime.

Step-by-Step Solution:
Step 1: For the pair (64, 23), compute the difference: 64 − 23 = 41. The number 41 has no divisors other than 1 and 41, so 41 is a prime number. Step 2: For the pair (42, 12), compute the difference: 42 − 12 = 30. The number 30 has divisors 1, 2, 3, 5, 6, 10, 15, and 30, so it is composite. Step 3: For the pair (63, 18), compute the difference: 63 − 18 = 45. The number 45 has divisors 1, 3, 5, 9, 15, and 45, so it is also composite. Step 4: For the pair (83, 20), compute the difference: 83 − 20 = 63. The number 63 has divisors 1, 3, 7, 9, 21, and 63, so it is composite. Step 5: For the extra option (55, 16), compute the difference: 55 − 16 = 39. The number 39 has divisors 1, 3, 13, and 39, so it is composite. Step 6: Only the difference in the pair (64, 23) is prime, while the differences in the other pairs are composite. Thus, (64, 23) is the odd pair.

Verification / Alternative check:
As a quick check, we can recall common prime numbers up to 50, such as 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47. Among the computed differences 41, 30, 45, 63, and 39, only 41 appears in this list. The remaining numbers are divisible by small primes like 2, 3, or 5, so they are clearly composite. This confirms that our classification is correct and that (64, 23) stands out as the only pair with a prime difference.

Why Other Options Are Wrong:
42, 12: The difference is 30, which is composite, so this pair belongs to the majority group.
63, 18: The difference is 45, again composite, matching the main pattern.
83, 20: The difference is 63, also composite and not exceptional.
55, 16: The difference is 39, which is composite, so this pair also fits the majority pattern.

Common Pitfalls:
A typical mistake is to mis calculate the difference or to mis classify numbers like 39 and 45 as prime when they are actually divisible by 3. Another pitfall is to look for a different pattern such as parity, which does not uniquely identify a single option here. To avoid these issues, always compute differences carefully and quickly factor them using small prime divisors such as 2, 3, 5, 7, and 11.

Final Answer:
The only pair whose difference is a prime number is (64, 23), so it is the odd one out.