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

(A) 97 (B) 65 (C) 35 (D) 13 (E) 5

Just want to point out an observation here: If you are short on time, you can eliminate 3 options in seconds and your probability of getting the right answer goes to 50%. (B), (D) and (E) can certainly not be the answer.

Here's why: Say, 65 is the answer (it is not a factor of n). But then at least one of 13 and 5 is not a factor of n because 65 = 13*5. Then there would be at least two correct answers but that is not possible. This means 65 is a factor of n and hence 13 and 5 both need to be factors of n too.
_________________

Bunuel, Do you have a list of problems like this. More specifically, problems that require me to break down exponents. I never see it until afterwards and then it makes perfect sense. I feel like need more practice.

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

(A) 97 (B) 65 (C) 35 (D) 13 (E) 5

We would never be asked to calculate 3^8 or 2^8, so we must approach this problem not as an arithmetic question but as an algebraic one.

The first thing we must recognize is that we are being tested on the algebraic factoring technique called the "difference of squares." Recall that the general form of the difference of squares is:

x^2 – y^2 = (x + y)(x – y)

Similarly, we can treat 3^8 – 2^8 as a difference of squares, which can be expressed as:

n = (3^4 + 2^4)(3^4 – 2^4)

We can further factor 3^4 – 2^4 as an additional difference of squares, which can be expressed as:

(3^2 + 2^2)(3^2 - 2^2)

This finally gives us:

n = 3^8 - 2^8 = (3^4 + 2^4)(3^2 + 2^2)(3^2 - 2^2)

The numbers are now easy to calculate:

n = (81 + 16)(9 + 4)(9 – 4)

n = (97)(13)(5)

We are being asked which of the answer choices is NOT a factor of n, which we have determined to be equal to the product (97)(13)(5). So we must find the answer choice that does not evenly divide into (97)(13)(5).

Right away we see that 97, 13 and 5 are all factors of (97)(13)(5).

This leaves us with 65 and 35. We should notice that (97)(13)(5) = (97)(65). Thus, 65 also is a factor of n. Only 35 is not.

Answer is C.
_________________

Scott Woodbury-Stewart Founder and CEO

GMAT Quant Self-Study Course 500+ lessons 3000+ practice problems 800+ HD solutions

Re: If n = 3^8 - 2^8, which of the following is NOT a factor of [#permalink]

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01 Apr 2017, 04:41

1

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n=3^8-2^8

Its a real time saver to learn algebraic identities. We know that: a^2-b^2= (a+b)(a-b)

Let's use it in the above expression. Why did I get this idea? Because 8= 2*4 n= [(3^4)^2- (2^4)^2] n= (3^4+2^4)(3^4-2^4) n=(81+16)(81-16) n= (97)(65) n=97*13*5

97,13, 5, and 65 are factors of n. The answer is C.
_________________

Help me make my explanation better by providing a logical feedback.

Re: If n = 3^8 - 2^8, which of the following is NOT a factor of [#permalink]

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14 Dec 2012, 03:19

Ans:

we will apply the formula a^2-b^2 here and expand the equation . in the end we get (3-2)(3+2)(3^2+2^2)(3^4+2^4)= 1x5x13x97 , therefore the answer is (C).
_________________

If n = 3^8 - 2^8, which of the following is NOT a factor of [#permalink]

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25 Nov 2014, 18:17

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Bunuel, Do you have a list of problems like this. More specifically, problems that require me to break down exponents. I never see it until afterwards and then it makes perfect sense. I feel like need more practice.

Then I would process elimination (A) 97 - 6305/97 is an integer. Not the answer (B) 65 - 6305/65 is an integer. Not the answer (C) 35 - 6305/35 is not an integer. The answer (D) 13 - 6305/13 is an integer. Not the answer (E) 5 - 6305/5 is an integer. Not the answer.