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If \(\mathrm{n} = 3^8 - 2^8\), which of the following is NOT a factor of \(\mathrm{n}\)?
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
Now here's the key insight: both \(\mathrm{3^8}\) and \(\mathrm{2^8}\) are perfect squares! Here's why:
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!
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:
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: \(\mathrm{n = 97 \times 5 \times 13}\)
Now we check each answer choice against our prime factorization \(\mathrm{n = 97 \times 5 \times 13}\):
Process Skill: APPLY CONSTRAINTS - Systematically checking each option against the complete factorization
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.
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.
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.
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.