In a certain numerical puzzle, the symbol # denotes a special operation between two numbers. It is given that 19 # 13 = 3, 25 # 3 = 11 and 36 # 10 = 13. Using the same rule, what is the value of 7 # 3 ?

Introduction / Context:
This problem is based on a coded numerical operation where the symbol # represents some hidden rule instead of a standard arithmetic operation. We are given three examples of input and output values involving this symbol and then asked to apply the same rule to a new pair of numbers. Such questions test the ability to identify simple patterns and relationships between numbers and then generalise that pattern to an unseen case without guessing randomly.

Given Data / Assumptions:

• 19 # 13 = 3.
• 25 # 3 = 11.
• 36 # 10 = 13.
• The same rule for # is used in all three examples.
• We are required to find the value of 7 # 3.
• Only basic arithmetic operations are assumed to underlie the pattern.

Concept / Approach:
The outputs 3, 11 and 13 are much smaller than the original numbers, which suggests that the operation is not simple multiplication or addition. A natural first idea is to look at the difference between the two numbers in each pair and then see how that difference might be transformed. For example, 19 - 13 is 6, and halving this gives 3, which matches the first result. We check whether halving the difference explains all given examples. If it does, then this becomes the working rule for the # operation and can be applied confidently to the new pair 7 and 3.

Step-by-Step Solution:
Step 1: For 19 # 13, compute the difference 19 - 13 = 6. Step 2: Halve this difference: 6 / 2 = 3, which matches the given result 3. Step 3: For 25 # 3, compute the difference 25 - 3 = 22. Step 4: Halve this difference: 22 / 2 = 11, which matches the second result 11. Step 5: For 36 # 10, compute the difference 36 - 10 = 26. Step 6: Again halving gives 26 / 2 = 13, which matches the third result 13. Step 7: The pattern is therefore clearly a # b = (a - b) / 2. Step 8: Apply the same rule to 7 # 3. The difference is 7 - 3 = 4. Step 9: Halve this difference: 4 / 2 = 2.

Verification / Alternative check:
All three original examples are perfectly explained by the single formula a # b = (a - b) / 2. There is no need to invent a more complicated rule involving digit sums or squares. If we tried such alternatives, they would quickly fail for at least one of the given pairs. Because a simple, consistent formula exists and matches every known case, it is overwhelmingly likely to be the intended rule, and using it for 7 and 3 is fully justified.

Why Other Options Are Wrong:
Option 21 corresponds to simply taking the product 7 * 3 and ignores the pattern of halving the difference. Option 26 and option 39 reflect other uncontrolled operations such as adding or multiplying the numbers without checking consistency with the examples. Option 10 neither comes from halving the difference nor matches any pattern demonstrated in the given data. Only 2 agrees with the clearly established formula.

Common Pitfalls:
A common error is to look at just one example and assume a rule like a - b or a / b without verifying against all given equations. Another frequent mistake is to jump to a complex rule involving digit sums or factorials, which is rarely required for simple exam questions. Always derive a candidate rule from one example and then rigorously test it against all other examples before trusting it. In this way, you avoid patterns that only fit partially and you arrive at the correct operation more reliably.

Final Answer:
Using the rule a # b = (a - b) / 2, the value of 7 # 3 is 2.