The sum and product of k positive integers are 12 and 81, respectively. If the k positive integers are all...

1. Translate the problem requirements: We have k positive integers that are all equal to each other. Their sum equals 12 and their product equals 81. We need to find the value of k.
2. Set up the mathematical relationship: Since all k integers are equal, let's call each integer 'x'. Then the sum is \(\mathrm{k} \times \mathrm{x} = 12\) and the product is \(\mathrm{x}^\mathrm{k} = 81\).
3. Express one variable in terms of the other: From the sum equation, we can find \(\mathrm{x} = \frac{12}{\mathrm{k}}\), then substitute this into the product equation.
4. Solve for k using the product equation: Substitute \(\mathrm{x} = \frac{12}{\mathrm{k}}\) into \(\mathrm{x}^\mathrm{k} = 81\) to get \(\left(\frac{12}{\mathrm{k}}\right)^\mathrm{k} = 81\), then test values of k to find which one works.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we know in plain English: We have some number of positive integers (let's call this number 'k'), and all these integers are exactly the same value. When we add all k integers together, we get 12. When we multiply all k integers together, we get 81.

For example, if we had 3 identical integers that each equal 4, then:

- The sum would be: \(4 + 4 + 4 = 12\)
- The product would be: \(4 \times 4 \times 4 = 64\)

But we need both the sum to equal 12 AND the product to equal 81. Our job is to find how many identical integers (k) we need.

Process Skill: TRANSLATE - Converting the word problem into mathematical understanding

2. Set up the mathematical relationship

Since all k integers are equal to each other, let's call each integer 'x'. Now we can express our requirements mathematically:

- Sum condition: If we have k integers, each equal to x, then \(\mathrm{x} + \mathrm{x} + \mathrm{x} + \ldots\) (k times) \(= \mathrm{k} \times \mathrm{x} = 12\)
- Product condition: If we multiply k integers, each equal to x, then \(\mathrm{x} \times \mathrm{x} \times \mathrm{x} \times \ldots\) (k times) \(= \mathrm{x}^\mathrm{k} = 81\)

So we have two equations:

- \(\mathrm{k} \times \mathrm{x} = 12\)
- \(\mathrm{x}^\mathrm{k} = 81\)

3. Express one variable in terms of the other

From the sum equation \(\mathrm{k} \times \mathrm{x} = 12\), we can solve for x:
\(\mathrm{x} = \frac{12}{\mathrm{k}}\)

This tells us that if we know k (how many integers we have), we can immediately find x (what each integer equals).

For instance:

- If \(\mathrm{k} = 3\), then \(\mathrm{x} = \frac{12}{3} = 4\)
- If \(\mathrm{k} = 4\), then \(\mathrm{x} = \frac{12}{4} = 3\)
- If \(\mathrm{k} = 6\), then \(\mathrm{x} = \frac{12}{6} = 2\)

4. Solve for k using the product equation

Now we substitute \(\mathrm{x} = \frac{12}{\mathrm{k}}\) into our product equation \(\mathrm{x}^\mathrm{k} = 81\):
\(\left(\frac{12}{\mathrm{k}}\right)^\mathrm{k} = 81\)

Rather than trying to solve this algebraically, let's test the answer choices systematically:

Testing k = 3:
\(\mathrm{x} = \frac{12}{3} = 4\)
Product check: \(4^3 = 64 \neq 81\) ✗

Testing k = 4:
\(\mathrm{x} = \frac{12}{4} = 3\)
Product check: \(3^4 = 81\) ✓
Sum check: \(4 \times 3 = 12\) ✓

Testing k = 5:
\(\mathrm{x} = \frac{12}{5} = 2.4\)
Product check: \((2.4)^5 = 79.6\ldots \neq 81\) ✗

Testing k = 6:
\(\mathrm{x} = \frac{12}{6} = 2\)
Product check: \(2^6 = 64 \neq 81\) ✗

Process Skill: APPLY CONSTRAINTS - Testing each possibility against both conditions

5. Final Answer

We found that \(\mathrm{k} = 4\) works perfectly. Let's verify:

- We have 4 identical positive integers
- Each integer equals 3
- Sum: \(3 + 3 + 3 + 3 = 12\) ✓
- Product: \(3 \times 3 \times 3 \times 3 = 81\) ✓

The answer is D. 4

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "all equal to each other"
Students might think this means the integers are distinct but form some pattern, rather than understanding that every single integer has exactly the same value. This leads to overcomplicating the setup with multiple variables instead of recognizing we only need one variable (x) for the common value.

2. Confusing sum and product relationships
Students may struggle to correctly translate "sum is 12" into \(\mathrm{k} \times \mathrm{x} = 12\) and "product is 81" into \(\mathrm{x}^\mathrm{k} = 81\). They might write these equations backwards or mix up which operation corresponds to which condition.

3. Not recognizing the constraint-testing strategy
Instead of setting up the substitution approach (\(\mathrm{x} = \frac{12}{\mathrm{k}}\) into \(\mathrm{x}^\mathrm{k} = 81\)), students might attempt complex algebraic manipulation or fail to see that testing answer choices systematically is the most efficient method.

Errors while executing the approach

1. Arithmetic errors in exponentiation
Students frequently make calculation errors when computing powers, especially with \(3^4 = 81\). They might incorrectly calculate \(3^4 = 64\) or \(4^3 = 81\), leading to selecting the wrong answer choice.

2. Forgetting to verify both conditions
When testing answer choices, students might check only the product condition (\(\mathrm{x}^\mathrm{k} = 81\)) and forget to verify that the sum condition (\(\mathrm{k} \times \mathrm{x} = 12\)) is also satisfied, or vice versa.

3. Incorrect substitution
When substituting \(\mathrm{x} = \frac{12}{\mathrm{k}}\) into the product equation, students might make algebraic errors such as writing \(\left(\frac{12}{\mathrm{k}}\right)^\mathrm{k}\) incorrectly or miscalculating the resulting values for each test case.

Errors while selecting the answer

1. Selecting based on partial verification
Students might find that \(\mathrm{k} = 6\) gives \(\mathrm{x} = 2\) and notice that \(2^6 = 64\) is close to 81, then select this answer without being precise about the exact match required.

2. Confusing k with x in the final answer
After finding that \(\mathrm{k} = 4\) and \(\mathrm{x} = 3\), students might mistakenly select answer choice E (3) because they're thinking of the value of each integer rather than the number of integers asked for in the question.