How many integers between 1 and 16, inclusive, have exactly 3 different positive integer factors? (Note: 6 is NOT such...

1. Translate the problem requirements: We need to find integers from 1 to 16 that have exactly 3 positive divisors (factors). The example shows 6 has 4 factors (1, 2, 3, 6), so it doesn't qualify.
2. Identify the mathematical pattern: Determine what type of numbers have exactly 3 factors by analyzing the relationship between a number's prime factorization and its factor count.
3. Apply the pattern systematically: Check each integer from 1 to 16 to see which ones fit the identified pattern.
4. Verify through factor counting: For candidate numbers, explicitly list their factors to confirm they have exactly 3 factors.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding exactly what we're looking for. We need integers between 1 and 16 (including both 1 and 16) that have exactly 3 positive integer factors.

What does "factors" or "divisors" mean? A factor of a number is any positive integer that divides evenly into that number. For example, the factors of 6 are 1, 2, 3, and 6 because:

• \(6 \div 1 = 6\) (no remainder)
• \(6 \div 2 = 3\) (no remainder)
• \(6 \div 3 = 2\) (no remainder)
• \(6 \div 6 = 1\) (no remainder)

Since 6 has 4 factors, it doesn't qualify for our search. We specifically need numbers with exactly 3 factors.

Process Skill: TRANSLATE - Converting the problem statement into clear mathematical understanding

2. Identify the mathematical pattern

Let's think about what kind of numbers could have exactly 3 factors. Let's examine some small numbers to see if we can spot a pattern:

• Number 1: Factors are just \(\{1\}\). That's 1 factor total.
• Number 2: Factors are \(\{1, 2\}\). That's 2 factors total.
• Number 3: Factors are \(\{1, 3\}\). That's 2 factors total.
• Number 4: Factors are \(\{1, 2, 4\}\). That's 3 factors total! ✓

Interesting! Let's check a few more:

• Number 8: Let's see... \(8 \div 1 = 8\), \(8 \div 2 = 4\), \(8 \div 4 = 2\), \(8 \div 8 = 1\). So factors are \(\{1, 2, 4, 8\}\). That's 4 factors.
• Number 9: Let's see... \(9 \div 1 = 9\), \(9 \div 3 = 3\), \(9 \div 9 = 1\). So factors are \(\{1, 3, 9\}\). That's 3 factors! ✓

Let me notice something: \(4 = 2^2\) and \(9 = 3^2\). Both are perfect squares of prime numbers!

When a number is the square of a prime (like \(\mathrm{p}^2\)), its factors are always exactly: \(\{1, \mathrm{p}, \mathrm{p}^2\}\). That's always exactly 3 factors.

Process Skill: INFER - Drawing the non-obvious conclusion that numbers with exactly 3 factors must be squares of primes

3. Apply the pattern systematically

Now I know I'm looking for squares of prime numbers between 1 and 16.

Let me list the prime numbers first: 2, 3, 5, 7, 11, 13, ...

Now let me square each prime and see which squares fall in our range [1, 16]:

• \(2^2 = 4\) ✓ (this is \(\leq 16\))
• \(3^2 = 9\) ✓ (this is \(\leq 16\))
• \(5^2 = 25\) ✗ (this is \(> 16\), so it's outside our range)
• \(7^2 = 49\) ✗ (this is \(> 16\), so it's outside our range)

Since \(5^2 = 25\) is already bigger than 16, all higher primes will give even larger squares.

So the only candidates are 4 and 9.

Process Skill: APPLY CONSTRAINTS - Systematically checking which prime squares fall within our given range

4. Verify through factor counting

Let me double-check by explicitly listing the factors:

For 4:

• \(4 \div 1 = 4\) ✓
• \(4 \div 2 = 2\) ✓
• \(4 \div 4 = 1\) ✓

Factors: \(\{1, 2, 4\}\) → Exactly 3 factors ✓

For 9:

• \(9 \div 1 = 9\) ✓
• \(9 \div 3 = 3\) ✓
• \(9 \div 9 = 1\) ✓

Factors: \(\{1, 3, 9\}\) → Exactly 3 factors ✓

Let me also verify that no other numbers work by checking a few more examples:

For 16 (which is \(2^4\), not a prime squared):

• \(16 \div 1 = 16\) ✓
• \(16 \div 2 = 8\) ✓
• \(16 \div 4 = 4\) ✓
• \(16 \div 8 = 2\) ✓
• \(16 \div 16 = 1\) ✓

Factors: \(\{1, 2, 4, 8, 16\}\) → 5 factors, so this doesn't work.

Perfect! Our analysis is confirmed.

5. Final Answer

We found exactly 2 integers between 1 and 16 (inclusive) that have exactly 3 different positive integer factors:

• \(4 = 2^2\) with factors \(\{1, 2, 4\}\)
• \(9 = 3^2\) with factors \(\{1, 3, 9\}\)

The answer is B. 2.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "exactly 3 factors" means
Students often confuse "factors" with "prime factors" or "composite factors." They might think the question is asking for numbers that can be written as a product of exactly 3 prime numbers (like \(2 \times 3 \times 5 = 30\)), rather than numbers that have exactly 3 divisors total. This leads them down completely wrong solution paths.

2. Failing to recognize the pattern that only squares of primes have exactly 3 factors
Many students will attempt to manually list factors for each number from 1 to 16 without recognizing the underlying mathematical pattern. While this brute-force approach can work, it's time-consuming and prone to errors. Students who don't identify that \(\mathrm{p}^2\) (where p is prime) always has exactly the factors \(\{1, \mathrm{p}, \mathrm{p}^2\}\) miss the elegant shortcut that makes this problem much simpler.

3. Incorrectly including 1 as a candidate
Some students might incorrectly think that 1 has exactly 3 factors, perhaps by confusing the concept or by incorrectly listing factors of 1. Since 1 only has itself as a factor, it should be eliminated early in the analysis, but students sometimes overlook this basic case.

Errors while executing the approach

1. Arithmetic errors when listing factors
When manually checking factors (especially for verification), students commonly miss factors or include non-factors. For example, when checking 12, they might forget that \(12 \div 6 = 2\), so 6 is also a factor, or they might incorrectly think that 5 divides 12. These computational mistakes lead to wrong factor counts.

2. Incorrectly calculating squares of primes
Students might make basic multiplication errors when computing \(\mathrm{p}^2\). For instance, they might calculate \(3^2\) as 6 instead of 9, or \(5^2\) as 20 instead of 25. Even small arithmetic mistakes here can lead to including wrong candidates or excluding correct ones.

3. Missing prime numbers or including composite numbers
When systematically checking squares of primes, students might forget that 2 is prime (since it's even), or they might incorrectly include squares of composite numbers like \(4^2 = 16\), thinking that 4 is prime. This leads to checking wrong candidates entirely.

Errors while selecting the answer

1. Counting the actual numbers instead of how many such numbers exist
Some students might arrive at the correct candidates (4 and 9) but then mistakenly select answer choice D (4) or E (9) instead of B (2). They confuse the question asking "how many integers" with the values of those integers themselves.

2. Including boundary cases incorrectly
Students might second-guess whether 1 and 16 should be included based on the phrase "between 1 and 16, inclusive." Even after correctly identifying 4 and 9 as the only valid numbers, they might add or subtract 1 from their count due to uncertainty about the inclusion of endpoints.