If n = 3^8 - 2^8, which of the following is NOT a factor of n?

1. Translate the problem requirements: We need to find which answer choice is NOT a factor of \(\mathrm{n = 3^8 - 2^8}\). This means we need to determine the prime factorization of n and check which given option doesn't divide n evenly.
2. Apply difference of squares factorization: Since both terms are perfect squares (\(\mathrm{3^8 = (3^4)^2}\) and \(\mathrm{2^8 = (2^4)^2}\)), use the difference of squares formula to break down the expression into manageable factors.
3. Continue factorization until reaching prime factors: Keep applying difference of squares and other factorization techniques to find all prime factors of n.
4. Test each answer choice for divisibility: Check whether each given option divides n by examining if all prime factors of the option are present in n's factorization.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're being asked to find. We have an expression \(\mathrm{n = 3^8 - 2^8}\), and we need to determine which of the five given options is NOT a factor of n.

Think of it this way: if a number is a factor of n, then when we divide n by that number, we get a whole number with no remainder. We need to find which option, when we try to divide n by it, will leave a remainder - meaning it doesn't divide evenly.

To do this systematically, we'll break down n into its prime factors (the building blocks), and then check if each answer choice can be built from those same building blocks.

Process Skill: TRANSLATE - Converting the factor question into a prime factorization problem

2. Apply difference of squares factorization

Now here's the key insight: both \(\mathrm{3^8}\) and \(\mathrm{2^8}\) are perfect squares! Here's why:

• \(\mathrm{3^8 = (3^4)^2 = 81^2}\)
• \(\mathrm{2^8 = (2^4)^2 = 16^2}\)

So our expression \(\mathrm{n = 3^8 - 2^8}\) becomes \(\mathrm{n = 81^2 - 16^2}\).

When we have something in the form \(\mathrm{a^2 - b^2}\), we can factor it as \(\mathrm{(a + b)(a - b)}\). This is like saying "the difference between two squares equals the sum times the difference."

So: \(\mathrm{n = 81^2 - 16^2 = (81 + 16)(81 - 16) = (97)(65)}\)

Let's verify: 81 + 16 = 97 and 81 - 16 = 65. Perfect!

3. Continue factorization until reaching prime factors

We now have \(\mathrm{n = 97 \times 65}\), but we're not done yet. We need to break these down further:

First, is 97 prime? Let's check if any small primes divide 97:

• 97 ÷ 2 = not divisible (97 is odd)
• 97 ÷ 3 = not divisible (9 + 7 = 16, not divisible by 3)
• 97 ÷ 5 = not divisible (doesn't end in 0 or 5)
• 97 ÷ 7 ≈ 13.86 (not whole)
• 97 ÷ 11 ≈ 8.82 (not whole)

Since √97 ≈ 9.8, we only need to check primes up to 9. So 97 is prime.

Now let's factor 65:
65 = 5 × 13 (both 5 and 13 are prime)

So our complete factorization is: \(\mathrm{n = 97 \times 5 \times 13}\)

4. Test each answer choice for divisibility

Now we check each answer choice against our prime factorization \(\mathrm{n = 97 \times 5 \times 13}\):

• (A) 97: This is one of our prime factors, so 97 IS a factor of n ✓
• (B) 65: We showed that \(\mathrm{65 = 5 \times 13}\), and both 5 and 13 are in our factorization, so 65 IS a factor of n ✓
• (C) 35: Let's check: \(\mathrm{35 = 5 \times 7}\). We have the factor 5 in our factorization, but we do NOT have the factor 7. Since 7 is not in our prime factorization, 35 is NOT a factor of n ✗
• (D) 13: This is one of our prime factors, so 13 IS a factor of n ✓
• (E) 5: This is one of our prime factors, so 5 IS a factor of n ✓

Process Skill: APPLY CONSTRAINTS - Systematically checking each option against the complete factorization

4. Final Answer

The answer is (C) 35.

To summarize our work: \(\mathrm{n = 3^8 - 2^8 = 97 \times 5 \times 13}\). Since \(\mathrm{35 = 5 \times 7}\), and the prime factor 7 does not appear in our factorization of n, 35 cannot be a factor of n. All other options (97, 65, 13, and 5) are indeed factors of n.

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Missing the difference of squares pattern
Students often fail to recognize that \(\mathrm{3^8 - 2^8}\) can be rewritten as \(\mathrm{(3^4)^2 - (2^4)^2}\), which is a difference of squares. Instead, they might try to calculate \(\mathrm{3^8}\) and \(\mathrm{2^8}\) separately (getting very large numbers like 6561 and 256) and then subtract, making the problem much more difficult than necessary.

Faltering Point 2: Stopping factorization too early
Even if students successfully apply the difference of squares to get \(\mathrm{n = 97 \times 65}\), they might stop here and try to test the answer choices directly against these two factors. They fail to realize that 65 can be further factored into \(\mathrm{5 \times 13}\), which is crucial for checking whether composite numbers like 35 are factors.

Errors while executing the approach

Faltering Point 1: Arithmetic errors in basic calculations
Students make simple calculation mistakes like 81 + 16 = 96 (instead of 97) or 81 - 16 = 64 (instead of 65). These errors propagate through the entire solution and lead to incorrect factorizations.

Faltering Point 2: Incorrect prime factorization of answer choices
When checking if answer choices are factors, students might incorrectly factor the given numbers. For example, they might think \(\mathrm{35 = 5 \times 6}\) instead of \(\mathrm{35 = 5 \times 7}\), or make errors in determining whether numbers like 97 are prime.

Errors while selecting the answer

Faltering Point 1: Misunderstanding the question requirement
The question asks for which option is NOT a factor, but students might select an option that IS a factor. This is a common error when students correctly identify all factors but then choose the wrong answer because they misread what the question is asking for.