An integer n that is greater than 1 is said to be "prime-saturated" if it has no prime factor greater...

1. Translate the problem requirements: An integer n is "prime-saturated" if ALL of its prime factors are strictly less than \(\sqrt{n}\). We need to find which answer choice satisfies this condition.
2. Calculate the square root threshold for each option: Determine \(\sqrt{n}\) for each answer choice to establish the upper limit that prime factors must stay below.
3. Find prime factorization for each candidate: Break down each number into its prime factors to identify what we're comparing against the threshold.
4. Apply the prime-saturated test: Check if every prime factor is strictly less than the calculated square root threshold.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what "prime-saturated" means in everyday language. Think of it this way: imagine you have a number, and you want to check if it's "prime-saturated." You need to do two things:

• Find all the prime numbers that multiply together to give you your original number (these are called prime factors)
• Calculate the square root of your original number
• Check that EVERY prime factor is smaller than that square root

If all prime factors are strictly less than \(\sqrt{n}\), then the number is prime-saturated. If even one prime factor is greater than or equal to \(\sqrt{n}\), then it's NOT prime-saturated.

2. Calculate the square root threshold for each option

Let's find the square root of each answer choice. This gives us the "threshold" that all prime factors must stay below:

1. \(\sqrt{6} \approx 2.45\)
2. \(\sqrt{35} \approx 5.92\)
3. \(\sqrt{46} \approx 6.78\)
4. \(\sqrt{66} \approx 8.12\)
5. \(\sqrt{75} \approx 8.66\)

Now we know exactly what each number's prime factors need to be smaller than.

3. Find prime factorization for each candidate

Let's break down each number into its prime factors:

1. \(6 = 2 \times 3\) (prime factors: 2 and 3)
2. \(35 = 5 \times 7\) (prime factors: 5 and 7)
3. \(46 = 2 \times 23\) (prime factors: 2 and 23)
4. \(66 = 2 \times 3 \times 11\) (prime factors: 2, 3, and 11)
5. \(75 = 3 \times 5^2 = 3 \times 5 \times 5\) (prime factors: 3 and 5)

Now we have everything we need to test each option.

4. Apply the prime-saturated test

Let's check each option systematically by comparing every prime factor to the square root threshold:

Option A) 6: Prime factors are 2 and 3, threshold is \(\sqrt{6} \approx 2.45\)
- Is \(2 < 2.45\)? ✓ Yes
- Is \(3 < 2.45\)? ✗ No (\(3 > 2.45\))
Not prime-saturated.

Option B) 35: Prime factors are 5 and 7, threshold is \(\sqrt{35} \approx 5.92\)
- Is \(5 < 5.92\)? ✓ Yes
- Is \(7 < 5.92\)? ✗ No (\(7 > 5.92\))
Not prime-saturated.

Option C) 46: Prime factors are 2 and 23, threshold is \(\sqrt{46} \approx 6.78\)
- Is \(2 < 6.78\)? ✓ Yes
- Is \(23 < 6.78\)? ✗ No (\(23 > 6.78\))
Not prime-saturated.

Option D) 66: Prime factors are 2, 3, and 11, threshold is \(\sqrt{66} \approx 8.12\)
- Is \(2 < 8.12\)? ✓ Yes
- Is \(3 < 8.12\)? ✓ Yes
- Is \(11 < 8.12\)? ✗ No (\(11 > 8.12\))
Not prime-saturated.

Option E) 75: Prime factors are 3 and 5, threshold is \(\sqrt{75} \approx 8.66\)
- Is \(3 < 8.66\)? ✓ Yes
- Is \(5 < 8.66\)? ✓ Yes
All prime factors are less than the threshold! This IS prime-saturated.

Final Answer

The answer is E) 75.

\(75 = 3 \times 5 \times 5\), and both prime factors (3 and 5) are strictly less than \(\sqrt{75} \approx 8.66\), making it the only prime-saturated number among the choices.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding "greater than or equal to": Students often miss that the definition says "no prime factor greater than or equal to \(\sqrt{n}\)" which means ALL prime factors must be STRICTLY LESS than \(\sqrt{n}\). They might incorrectly think a prime factor can equal \(\sqrt{n}\) and still satisfy the condition.

2. Confusing the direction of comparison: Students may flip the inequality and think they need to find numbers where prime factors are greater than \(\sqrt{n}\) instead of less than \(\sqrt{n}\), completely reversing the prime-saturated condition.

3. Misinterpreting what constitutes a prime factor: Students might include composite factors or forget that repeated prime factors (like 5 appearing twice in \(75 = 3 \times 5^2\)) only need to be checked once since we only care about distinct prime factors.

Errors while executing the approach

1. Prime factorization errors: Students often make mistakes when breaking down numbers into prime factors, especially for numbers like \(46 = 2 \times 23\) where they might not recognize that 23 is prime, or incorrectly factorize 75 as something other than \(3 \times 5^2\).

2. Square root calculation mistakes: Students may incorrectly estimate square roots, particularly for numbers like \(\sqrt{35} \approx 5.92\) or \(\sqrt{75} \approx 8.66\), leading to wrong threshold values for comparison.

3. Incomplete checking: Students might check only some prime factors against the square root threshold but forget to verify ALL prime factors, especially when a number has multiple prime factors like \(66 = 2 \times 3 \times 11\).

Errors while selecting the answer

1. Stopping at the first "close" answer: Students might find that option A) or B) seems to work without carefully checking all calculations, and select it without verifying all five options systematically.

2. Boundary confusion: When a prime factor is very close to the square root threshold (like checking if \(7 < 5.92\)), students might incorrectly conclude it satisfies the condition due to hasty comparison or rounding errors.