In a certain numerical code, the symbol * represents a special operation on two numbers. If 5 * 3 is defined to be equal to 19 and 8 * 5 is defined to be equal to 49 using the same hidden rule, then according to this pattern what should be the value of 6 * 4 ?

Introduction / Context:
This question belongs to the coding and decoding category where a normal arithmetic symbol is used with a hidden rule instead of its usual meaning. The operation * does not represent ordinary multiplication but a specially designed pattern. To find the value of 6 * 4, we must first decode the pattern from the two examples 5 * 3 = 19 and 8 * 5 = 49 and then apply the same logic consistently. Such questions test pattern recognition, algebraic thinking and the ability to generalise from given examples without guessing randomly.

Given Data / Assumptions:

• The coded operation is written as a * b where a and b are numbers.
• According to the given rule, 5 * 3 = 19.
• According to the same rule, 8 * 5 = 49.
• We assume the same single rule works consistently for all pairs.
• We need to compute 6 * 4 using this hidden rule.

Concept / Approach:
In coding operation problems, we usually start by checking combinations of standard arithmetic operations such as addition, subtraction, multiplication and squares. Here, normal multiplication 5 * 3 = 15 is close to 19, and 8 * 5 = 40 is close to 49, so it is natural to check what extra quantity is added. The differences 19 - 15 = 4 and 49 - 40 = 9 are perfect squares, which strongly suggests that the rule adds the square of the difference between the two numbers. So we try the pattern a * b = a * b + (a - b)^2 and verify it with the given data before using it for the unknown case.

Step-by-Step Solution:
Step 1: Compute the ordinary product for the first pair: 5 * 3 (normal) = 15. Step 2: Compute the difference between the numbers: 5 - 3 = 2 and its square: 2^2 = 4. Step 3: Add this square to the product: 15 + 4 = 19, which matches the given value. Step 4: Check the second pair using the same rule. Normal product: 8 * 5 = 40. Step 5: Difference: 8 - 5 = 3, square: 3^2 = 9, so 40 + 9 = 49, again matching the given value. Step 6: Now apply the same rule to 6 and 4. Normal product: 6 * 4 = 24. Step 7: Difference: 6 - 4 = 2, square: 2^2 = 4, so coded value = 24 + 4 = 28.

Verification / Alternative check:
We have already verified the proposed rule with both of the given examples, which is the best way to confirm that the pattern is reliable. The same expression a * b = a * b + (a - b)^2 works perfectly for (5, 3) and for (8, 5). No simpler rule using only addition or only subtraction fits both pairs at the same time. This strongly indicates that the intended pattern is to take the product and then add the square of the difference between the two numbers. Once that is confirmed, using it for 6 and 4 is straightforward and gives 28 as the unique consistent value.

Why Other Options Are Wrong:
Option 24 corresponds to simply multiplying 6 by 4 without applying any coding rule, so it ignores the additional square term and does not match the pattern used in the examples. Option 18 does not result from any natural combination of product and difference that also works for the two given cases, so it breaks consistency. Option 16 would match the square of 4, but there is no consistent way to justify that using the given coded examples. Option 32 does not match the verified formula either, because that would require adding 8 instead of 4, which is not supported by the previous data. Therefore these options are deliberately misleading distractors.

Common Pitfalls:
Many learners either assume that * is still ordinary multiplication or they pick a rule that fits only one example but fails on the second pair. A frequent error is to add or subtract the numbers directly without squaring the difference, which cannot satisfy both given equations. Another mistake is to guess the answer without checking that the same pattern holds for every example in the question. In coding and decoding problems, always ensure that the rule is consistent with all given relations before applying it to find the unknown value.

Final Answer:
Using the pattern a * b = a * b + (a - b)^2, the coded value of 6 * 4 is 28.