In the following number pair analogy, 534 is related to 2. Using the same digit based pattern, which of the following pairs correctly completes the analogy ? : ?

Introduction / Context:
This question involves a relation between a three digit number and a single digit number based on the nature of its digits. The pair 534 and 2 at first seems mysterious, but by looking at the properties of the digits 5, 3, and 4 we can discover a simple rule. Once we identify this rule, we must apply it to the option pairs and find which pair follows the same digit based relationship.

Given Data / Assumptions:

• Given pair: 534 : 2
• Option pairs: 753 : 3, 421 : 2, 268 : 2, 468 : 1.
• We consider prime and composite nature of digits within each three digit number.

Concept / Approach:
Let us inspect the digits of 534. The digits are 5, 3, and 4. Among these, 5 and 3 are prime numbers, while 4 is composite. The count of prime digits is 2, which matches the second number in the pair. This suggests that the rule may be "second number equals the number of prime digits in the first number". We then check each option to see which pair obeys this rule.

Step-by-Step Solution:
Step 1: Analyse 534. Digits are 5, 3, and 4. Step 2: Determine which digits are prime. 5 is prime, 3 is prime, and 4 is not prime. Step 3: Count the prime digits. There are 2 prime digits, which matches the related number 2 in the pair 534 : 2. Step 4: Apply this rule to the option 753. Digits are 7, 5, and 3, and all three are prime. Step 5: Count the prime digits in 753. Since 7, 5, and 3 are all prime, the count is 3, so 753 should be related to 3. Step 6: Observe that the option 753 : 3 exactly reflects this relationship.

Verification / Alternative check:
Let us verify the rule for the remaining options. For 421, digits are 4, 2, and 1. Among these only 2 is prime, so the count of prime digits is 1, not 2. Therefore 421 : 2 does not satisfy the rule. For 268, digits are 2, 6, and 8. Only 2 is prime, so the count is 1, not 2. For 468, digits 4, 6, and 8 are all composite, so the count of prime digits is 0, not 1. Hence none of those pairs maintain the same pattern. Only 753 : 3 is consistent with the mapping rule derived from 534 : 2.

Why Other Options Are Wrong:
421 : 2 would be correct only if there were exactly two prime digits, but 4 and 1 are not prime. 268 : 2 again suggests two prime digits while there is only one. 468 : 1 claims one prime digit but actually has no prime digits at all. Since the pattern requires an exact count of prime digits, these mismatches show that the alternatives are incorrect.

Common Pitfalls:
A common difficulty is remembering that 1 is not considered a prime number and that prime digits are only 2, 3, 5, and 7. Students sometimes mistakenly treat 1 as prime and miscount. Another pitfall is trying operations like sums or products of digits without first exploring simpler digit properties such as parity or primality. In digit based analogies, checking how many digits share a particular property is often a useful technique.

Final Answer:
The pair that correctly completes the analogy 534 : 2 :: ? : ? is 753 : 3.