A number is said to be prime saturated if the product of all the different positive prime factors of n...

1. Translate the problem requirements: A prime saturated number means the product of all its distinct prime factors must be less than the square root of the number itself. We need to find the largest two-digit number with this property.
2. Establish the working constraint: Since we want the greatest two-digit prime saturated integer, we'll work backwards from 99 and test each number systematically until we find one that satisfies the condition.
3. Apply the prime saturation test efficiently: For each candidate number, find its prime factorization, multiply the distinct prime factors, and compare this product to the square root of the number.
4. Verify the solution: Once we identify a prime saturated number, confirm it's the largest by showing that larger two-digit numbers fail the prime saturation test.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what "prime saturated" means in everyday language. A number is prime saturated when the product of all its different prime factors is smaller than the square root of the number itself.

For example, let's say we have the number 12:

• Prime factorization of 12: \(2^2 \times 3 = 4 \times 3\)
• Different prime factors: 2 and 3
• Product of different prime factors: \(2 \times 3 = 6\)
• Square root of 12: \(\sqrt{12} \approx 3.46\)
• Since \(6 > 3.46\), the number 12 is NOT prime saturated

We need to find the largest two-digit number (between 10 and 99) that satisfies this prime saturation condition.

Process Skill: TRANSLATE - Converting the mathematical definition into testable criteria

2. Establish the working constraint

Since we want the greatest two-digit prime saturated integer, we'll work backwards from 99. This makes sense because once we find a prime saturated number, we only need to verify that larger numbers don't work.

Our approach will be:

• Start with 99 and work downward
• For each number, find its prime factorization
• Calculate the product of distinct prime factors
• Compare this product to the square root
• Stop when we find the first (largest) number that works

3. Apply the prime saturation test efficiently

Let's test the numbers systematically:

Testing 99:

• \(99 = 3^2 \times 11 = 9 \times 11\)
• Different prime factors: 3 and 11
• Product: \(3 \times 11 = 33\)
• \(\sqrt{99} \approx 9.95\)
• Since \(33 > 9.95\), 99 is NOT prime saturated

Testing 98:

• \(98 = 2 \times 49 = 2 \times 7^2\)
• Different prime factors: 2 and 7
• Product: \(2 \times 7 = 14\)
• \(\sqrt{98} \approx 9.90\)
• Since \(14 > 9.90\), 98 is NOT prime saturated

Testing 97:

• 97 is prime (only prime factor is 97 itself)
• Product of different prime factors: 97
• \(\sqrt{97} \approx 9.85\)
• Since \(97 > 9.85\), 97 is NOT prime saturated

Testing 96:

• \(96 = 32 \times 3 = 2^5 \times 3\)
• Different prime factors: 2 and 3
• Product: \(2 \times 3 = 6\)
• \(\sqrt{96} \approx 9.80\)
• Since \(6 < 9.80\), 96 IS prime saturated! ✓

Process Skill: APPLY CONSTRAINTS - Systematically testing the condition while working backwards from the maximum

4. Verify the solution

We found that 96 is prime saturated because:

• \(96 = 2^5 \times 3\)
• Product of distinct prime factors = \(2 \times 3 = 6\)
• \(\sqrt{96} \approx 9.80\)
• \(6 < 9.80\) ✓

We also verified that 97, 98, and 99 are all NOT prime saturated, which confirms that 96 is indeed the largest two-digit prime saturated integer.

Looking at our answer choices, 96 corresponds to choice D.

Final Answer

The greatest two-digit prime saturated integer is 96.

Answer: D

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the definition of "prime saturated"

Students may confuse the product of all prime factors (including repeated ones) with the product of distinct prime factors only. For example, with \(96 = 2^5 \times 3\), they might calculate \(2\times2\times2\times2\times2\times3 = 96\) instead of just \(2\times3 = 6\). This fundamental misreading of the definition would lead to incorrect calculations throughout.

2. Incorrect inequality direction

Students may flip the inequality and look for numbers where the product of prime factors is greater than the square root, instead of less than. This would lead them to conclude that 99, 98, and 97 are prime saturated (since \(33 > 9.95\), \(14 > 9.90\), \(97 > 9.85\)), making them select 99 as the answer.

3. Starting from the smallest two-digit number instead of working backwards

Since the question asks for the "greatest" two-digit prime saturated integer, students might start testing from 10 upwards instead of from 99 downwards. This approach is inefficient and may cause them to stop at the first large prime saturated number they find, potentially missing the actual maximum.

Errors while executing the approach

1. Incorrect prime factorization

Students may make errors in breaking down numbers into prime factors. For example, they might incorrectly factor 96 as \(2^4 \times 6\) instead of \(2^5 \times 3\), leading to the wrong distinct prime factors. This would give them a product of \(2\times6 = 12\) instead of \(2\times3 = 6\).

2. Computational errors with square roots

Students may incorrectly estimate square roots or use imprecise values. For instance, they might approximate \(\sqrt{96}\) as 10 instead of 9.80, which could lead to wrong conclusions about whether the inequality \(6 < \sqrt{96}\) holds true.

3. Forgetting to test prime numbers correctly

When testing prime numbers like 97, students might forget that the only prime factor is the number itself, so the "product of distinct prime factors" is just 97. They might incorrectly think that prime numbers automatically satisfy the condition or handle them as special cases incorrectly.

Errors while selecting the answer

1. Selecting the first prime saturated number found instead of the greatest

If students work upwards or test numbers randomly, they might find that a smaller number like 90 (\(2\times3\times5 = 30\), \(\sqrt{90} \approx 9.5\), but \(30 > 9.5\) so this doesn't work) or some other prime saturated number and select it without continuing to verify it's the maximum possible.

2. Mismatching the calculated answer with the answer choices

Students might correctly determine that 96 is prime saturated but then incorrectly map this to the wrong letter choice, perhaps selecting C (97) instead of D (96) due to careless reading of the options.