In this number analogy, 76 is related to 42 by a simple digit operation. Using the same operation, what number should replace the question mark in 46 : ?

Introduction / Context:
This problem is a digit based number analogy that focuses on the relationship between a two digit number and a second number formed from its digits. The given pair 76 : 42 suggests that there is some arithmetic operation applied to the digits 7 and 6 to obtain 42. Your task is to identify that operation and then apply it in the same way to the pair 46 : ? to find the missing term from the options. Such questions test your ability to quickly experiment with sums, differences and products of digits.

Given Data / Assumptions:

The first known pair is 76 : 42.
The second pair is 46 : ? with the answer to be found.
We work with the individual digits of the first number, not with any advanced number theory concept.
Options provided are 52, 24, 36 and 64.
We assume a simple operation like addition, subtraction, multiplication or a combination of digits is used consistently.

Concept / Approach:
A natural way to proceed is to break 76 into its digits 7 and 6 and test common operations. The sum 7 + 6 equals 13, which does not match 42. The difference 7 - 6 equals 1, which also does not match. The product 7 * 6 equals 42, which directly matches the right hand side. This suggests that the relationship is: second number equals product of the digits of the first number. Once this rule is established, we apply the same idea to 46 by multiplying 4 and 6.

Step-by-Step Solution:
Step 1: Split 76 into digits 7 and 6. Step 2: Compute 7 + 6 = 13. This does not give 42, so simple addition is not the rule. Step 3: Compute 7 - 6 = 1. This also does not match 42, so the rule is not subtraction. Step 4: Compute 7 * 6 = 42. This exactly matches the paired value, so the rule is product of digits. Step 5: Now apply the same rule to 46. Split 46 into digits 4 and 6. Step 6: Compute 4 * 6 = 24. Step 7: Conclude that 46 should be related to 24 in the same way that 76 is related to 42.

Verification / Alternative check:
To verify, consider whether any other plausible digit operation could produce 42 from 76 in a neat, exam friendly manner. Concatenation or reversal do not help, and neither do more complicated operations like 7^2 + 6 or similar, because they would not generalise cleanly to 46. Multiplying the digits gives an exact and simple result. Repeating this process for 46 gives 4 * 6 = 24, which appears in the options. None of the other options 52, 36 or 64 can be reached by such a simple digits product. This consistency strongly supports the product rule and the answer 24.

Why Other Options Are Wrong:
52 does not result from any basic digit operation on 4 and 6 such as sum, difference or product. Thirty six would be 6 * 6, which would require two identical digits, unlike the pair 4 and 6. Sixty four could arise from 4^3 but does not match a symmetric operation that would also work cleanly on 76. In contrast, 24 arises directly from 4 * 6, mirroring how 42 arises from 7 * 6, and so it is the only choice that preserves the analogy.

Common Pitfalls:
Many candidates initially try to interpret the right hand side as a new two digit number made by rearranging the original digits or adding an extra step. In fact, the simplest approach is usually correct in such problems. Forgetting to test multiplication after trying only sums and differences is a frequent oversight. To avoid this, always go through the standard list of operations on digits systematically: sum, difference, product and sometimes quotient. This disciplined pattern search quickly reveals the correct rule here.

Final Answer:
Since 42 is the product 7 * 6 and 24 is the product 4 * 6, the correct completion of the analogy is 24.