If integer p is a positive factor of 36 and integer t is a positive factor of 27, which of...

1. Translate the problem requirements: We need to find which product CANNOT equal \(\mathrm{p}^2 - \mathrm{t}^2\) where p is a positive factor of 36 and t is a positive factor of 27. This means we need to identify all possible values that \(\mathrm{p}^2 - \mathrm{t}^2\) can take, then see which answer choice is impossible.
2. Identify all possible factor values: List all positive factors of 36 for p and all positive factors of 27 for t to establish our working set of values.
3. Apply difference of squares pattern recognition: Since \(\mathrm{p}^2 - \mathrm{t}^2 = (\mathrm{p}+\mathrm{t})(\mathrm{p}-\mathrm{t})\), we need to find which products can be expressed in this form using our available factor values.
4. Systematic elimination through strategic checking: Rather than calculating every possible \(\mathrm{p}^2 - \mathrm{t}^2\), use the factored form to check if each answer choice can be written as \((\mathrm{p}+\mathrm{t})(\mathrm{p}-\mathrm{t})\) for some valid p and t values.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we're actually looking for here. We have two integers: p (which must be a positive factor of 36) and t (which must be a positive factor of 27). We need to find which of the given products CANNOT equal \(\mathrm{p}^2 - \mathrm{t}^2\).

This means we're going to figure out all the possible values that \(\mathrm{p}^2 - \mathrm{t}^2\) can take, and then check which answer choice is impossible to achieve.

Think of it like this: we have a limited menu of values for p and t, so we can only create a limited menu of results for \(\mathrm{p}^2 - \mathrm{t}^2\).

Process Skill: TRANSLATE - Converting the problem from words into a clear mathematical goal

2. Identify all possible factor values

Let's find our complete menu of values. For p, we need all positive factors of 36:

\(36 = 2^2 \times 3^2\), so the positive factors are: 1, 2, 3, 4, 6, 9, 12, 18, 36

For t, we need all positive factors of 27:

\(27 = 3^3\), so the positive factors are: 1, 3, 9, 27

Now we have our complete sets: p can be any of {1, 2, 3, 4, 6, 9, 12, 18, 36} and t can be any of {1, 3, 9, 27}.

3. Apply difference of squares pattern recognition

Here's the key insight: instead of calculating \(\mathrm{p}^2 - \mathrm{t}^2\) for every single combination (which would be tedious), let's use an algebraic pattern.

We know that \(\mathrm{p}^2 - \mathrm{t}^2\) can be factored as \((\mathrm{p} + \mathrm{t})(\mathrm{p} - \mathrm{t})\). This means any value that \(\mathrm{p}^2 - \mathrm{t}^2\) can equal must be expressible as a product of two factors where:

• The first factor is \((\mathrm{p} + \mathrm{t})\) for some valid p and t
• The second factor is \((\mathrm{p} - \mathrm{t})\) for the same p and t
• Note that \(\mathrm{p} + \mathrm{t} > \mathrm{p} - \mathrm{t}\) always, and both must have the same parity (both odd or both even)

Process Skill: VISUALIZE - Seeing the algebraic structure rather than brute force calculation

4. Systematic elimination through strategic checking

Let's check each answer choice to see if it can be written as \((\mathrm{p} + \mathrm{t})(\mathrm{p} - \mathrm{t})\):

(A) (12)(6) = 72: Can this equal \((\mathrm{p} + \mathrm{t})(\mathrm{p} - \mathrm{t})\) where \(\mathrm{p} + \mathrm{t} = 12\) and \(\mathrm{p} - \mathrm{t} = 6\)?
Solving: p = 9, t = 3. Check: 9 is a factor of 36 ✓, 3 is a factor of 27 ✓
Verification: \(9^2 - 3^2 = 81 - 9 = 72\) ✓

(B) (19)(17) = 323: Can this equal \((\mathrm{p} + \mathrm{t})(\mathrm{p} - \mathrm{t})\) where \(\mathrm{p} + \mathrm{t} = 19\) and \(\mathrm{p} - \mathrm{t} = 17\)?
Solving: p = 18, t = 1. Check: 18 is a factor of 36 ✓, 1 is a factor of 27 ✓
Verification: \(18^2 - 1^2 = 324 - 1 = 323\) ✓

(C) (39)(13) = 507: Can this equal \((\mathrm{p} + \mathrm{t})(\mathrm{p} - \mathrm{t})\) where \(\mathrm{p} + \mathrm{t} = 39\) and \(\mathrm{p} - \mathrm{t} = 13\)?
Solving: p = 26, t = 13. Check: Is 26 a factor of 36? No! ✗
Since 26 is not in our list of factors of 36, this is impossible.

(D) (45)(27) = 1215: Can this equal \((\mathrm{p} + \mathrm{t})(\mathrm{p} - \mathrm{t})\) where \(\mathrm{p} + \mathrm{t} = 45\) and \(\mathrm{p} - \mathrm{t} = 27\)?
Solving: p = 36, t = 9. Check: 36 is a factor of 36 ✓, 9 is a factor of 27 ✓

(E) (63)(9) = 567: Can this equal \((\mathrm{p} + \mathrm{t})(\mathrm{p} - \mathrm{t})\) where \(\mathrm{p} + \mathrm{t} = 63\) and \(\mathrm{p} - \mathrm{t} = 9\)?
Solving: p = 36, t = 27. Check: 36 is a factor of 36 ✓, 27 is a factor of 27 ✓

Process Skill: APPLY CONSTRAINTS - Using the factor restrictions to eliminate impossible cases

4. Final Answer

The answer is (C) (39)(13).

This product cannot equal \(\mathrm{p}^2 - \mathrm{t}^2\) because it would require p = 26, but 26 is not a positive factor of 36. All other answer choices can be achieved with valid factor combinations.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the "CANNOT" requirement: Students often rush and look for which product CAN equal \(\mathrm{p}^2 - \mathrm{t}^2\), instead of identifying which one CANNOT. This leads them to find valid combinations and incorrectly eliminate viable answer choices.

2. Missing the difference of squares factorization insight: Many students attempt to calculate \(\mathrm{p}^2 - \mathrm{t}^2\) for all possible combinations of p and t values (9 × 4 = 36 calculations) instead of recognizing that \(\mathrm{p}^2 - \mathrm{t}^2 = (\mathrm{p} + \mathrm{t})(\mathrm{p} - \mathrm{t})\). This makes the problem much more time-consuming and increases calculation errors.

3. Overlooking constraint restrictions: Students may forget that p must be a factor of 36 AND t must be a factor of 27 simultaneously. They might check if numbers are factors of either 36 OR 27, rather than ensuring both conditions are met for the same combination.

Errors while executing the approach

1. Incorrect factor identification: Students commonly make errors when listing factors. For \(36 = 2^2 \times 3^2\), they might miss factors like 18 or incorrectly include non-factors. For \(27 = 3^3\), they might forget that 27 itself is a factor or incorrectly include \(9^2 = 81\).

2. Solving the system of equations incorrectly: When trying to find p and t from \((\mathrm{p} + \mathrm{t})(\mathrm{p} - \mathrm{t})\), students often make algebraic mistakes. For example, with \(\mathrm{p} + \mathrm{t} = 39\) and \(\mathrm{p} - \mathrm{t} = 13\), they might get p = 25 or p = 28 instead of the correct p = 26, leading to wrong conclusions about factor validity.

3. Verification calculation errors: Students may correctly identify the values of p and t but then make arithmetic errors when verifying \(\mathrm{p}^2 - \mathrm{t}^2\). For instance, calculating \(18^2 - 1^2\) as 325 instead of 323, or \(36^2 - 27^2\) incorrectly, leading them to think a valid combination doesn't work.

Errors while selecting the answer

1. Selecting the first invalid case found: Students might find that answer choice (C) requires p = 26 (not a factor of 36) and immediately select it without checking if other answer choices might also be impossible. While this leads to the correct answer in this case, it's not a thorough approach.

2. Confusing the constraint violation: Students may correctly calculate that (39)(13) requires p = 26 and t = 13, but then check if 13 is a factor of 36 instead of checking if 26 is a factor of 36, leading to incorrect elimination.