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Manager  Joined: 02 Dec 2012
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If n = 3^8 - 2^8, which of the following is NOT a factor of  [#permalink]

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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
Math Expert V
Joined: 02 Sep 2009
Posts: 58446
Re: If n = 3^8 - 2^8, which of the following is NOT a factor of  [#permalink]

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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

Apply $$a^2-b^2=(a-b)(a+b)$$:

$$n = 3^8 - 2^8=(3^4-2^4)(3^4+2^4)=65*97=5*13*97$$ --> 7 is not a factor of n, therefore 35=5*7 also is not a factor of n.

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Intern  Joined: 24 Apr 2012
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Re: If n = 3^8 - 2^8, which of the following is NOT a factor of  [#permalink]

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

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1
3^8 - 2^8
=(3^4 + 2^4)(3^4 - 2^4)
=(3^4 + 2^4)(3^2 + 2^2)(3^2 - 2^2)
=(3^4 + 2^4)(3^2 + 2^2)(3 + 2)(3 - 2)
=(81 + 16)(9 + 4)(3 + 2)(3 - 2)
=97*13*5*1

A. 97 -- Factor of the expression
B. 65 -- Factor of the expression
C. 35
D. 13 -- Factor of the expression
E. 5 -- Factor of the expression

Correct answer is C.
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Re: If n = 3^8 – 2^8 which of the following is NOT a factor of n  [#permalink]

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(3^8-2^8)
=(3-2)(3+2)(3^2+2^2)(3^4+2^4)
=1*5*13*97
35 is not a factor....so C
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Re: If n = 3^8 - 2^8, which of the following is NOT a factor of  [#permalink]

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2
$$3^8 - 2^8$$

$$= 81^2 - 16^2$$

= 97 * 65

35 is not a factor; all other options stand fit

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

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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

3^8=81^2
2^8 = 16^2

now 81^2-16^2 in the format a^2-b^2 = (a+b)(a-b)

so 97*65

from the options only 35 is not a factor of above value
Intern  Joined: 17 Aug 2014
Posts: 2
If n = 3^8 - 2^8, which of the following is NOT a factor of  [#permalink]

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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.
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Posts: 58446
Re: If n = 3^8 - 2^8, which of the following is NOT a factor of  [#permalink]

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keysx015 wrote:
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.

Search the pages below.
DS Divisibility/Multiples/Factors questions to practice: search.php?search_id=tag&tag_id=354
PS Divisibility/Multiples/Factors questions to practice: search.php?search_id=tag&tag_id=185
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Senior Manager  Joined: 20 Aug 2015
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Location: India
GMAT 1: 760 Q50 V44 If n = 3^8 - 2^8, which of the following is NOT a factor of  [#permalink]

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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

Whenever you see anything of the form $$a^2 - b^2$$, write it as (a-b)*(a+b)

Coming to the question at hand,
n = $$3^8 - 2^8$$ =$$(3^4 - 2^4) * (3^4 + 2^4)$$

Again applying the same rule on $$(3^4 - 2^4)$$ ,
n = $$(3^2 - 2^2)*(3^2 + 2^2)*(3^4 + 2^4)$$ = $$(3 - 2)*(3+2)*(3^2 + 2^2)*(3^4 + 2^4)$$
n = 1*5*(9+4)*(81+16) = 1*5*13*97

On checking the options, we see that 35 cannot be formed by the factors of n, hence the correct answer

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

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

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2
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.

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Manager  S
Joined: 03 Jan 2017
Posts: 134
Re: If n = 3^8 - 2^8, which of the following is NOT a factor of  [#permalink]

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n=3^8-2^8
let's split it
(3^4-2^4)*(3^4+2^4)=65*97
Then let's test the answers
C is fine
Manager  B
Joined: 06 Dec 2016
Posts: 231
Re: If n = 3^8 - 2^8, which of the following is NOT a factor of  [#permalink]

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My approach

3^8 - 2^8
(3^4)^2 - (2^4)^2
(81)^2 - (16)^2
6561 - 256
6305

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.
Director  G
Joined: 02 Sep 2016
Posts: 649
Re: If n = 3^8 - 2^8, which of the following is NOT a factor of  [#permalink]

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

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VeritasPrepKarishma wrote:
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.

Responding to a pm:
Check here for the answer: https://gmatclub.com/forum/if-n-3-8-2-8 ... l#p1154050

What I have given above is the way to eliminate 3 options. That gives you a 50-50 probability of getting the answer (up from 20%). To get to the answer, we need to evaluate n (as done by Bunuel above).
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If n = 3^8 - 2^8, which of the following is NOT a factor of  [#permalink]

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Bunuel wrote:
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

Apply $$a^2-b^2=(a-b)(a+b)$$:

$$n = 3^8 - 2^8=(3^4-2^4)(3^4+2^4)=65*97=5*13*97$$ --> 7 is not a factor of n, therefore 35=5*7 also is not a factor of n.

niks18 hello, can you please give me a prompt how to solve this question i got stuck ...

$$n = 3^8 - 2^8=(3^4-2^4)(3^4+2^4)$$ from here i get this $$9^8+6^8-6^8-4^8$$ now what next ? by the way, my first thought looking at this $$n = 3^8 - 2^8$$, was that I simply need to subtract so I got simply 1 and got confused... why wouldn't this approch work here ... then I saw Bunuel s mind blowing approach so started solving and got stuck half way or may be on 1/3 of the way by the way this formula $$a^2-b^2=(a-b)(a+b)$$ features exponent 2 where as in the question it is 8, so no materr what is exponent this formula would worki right ? Retired Moderator D
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Re: If n = 3^8 - 2^8, which of the following is NOT a factor of  [#permalink]

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dave13 wrote:
Bunuel wrote:
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

Apply $$a^2-b^2=(a-b)(a+b)$$:

$$n = 3^8 - 2^8=(3^4-2^4)(3^4+2^4)=65*97=5*13*97$$ --> 7 is not a factor of n, therefore 35=5*7 also is not a factor of n.

niks18 hello, can you please give me a prompt how to solve this question i got stuck ...

$$n = 3^8 - 2^8=(3^4-2^4)(3^4+2^4)$$ from here i get this $$9^8+6^8-6^8-4^8$$ now what next ? by the way, my first thought looking at this $$n = 3^8 - 2^8$$, was that I simply need to subtract so I got simply 1 and got confused... why wouldn't this approch work here ... then I saw Bunuel s mind blowing approach so started solving and got stuck half way or may be on 1/3 of the way by the way this formula $$a^2-b^2=(a-b)(a+b)$$ features exponent 2 where as in the question it is 8, so no materr what is exponent this formula would worki right ? Hi dave13

The highlighted part is incorrect. Go through multiplication of exponents once to understand your mistake there.

Secondly $$a^2-b^2=(a-b)(a+b)$$. now we are given $$3^8 - 2^8$$ whose exponents are 8, so to use the formula you can manupulate the exponents as

$$(3^4)^2 - (2^4)^2$$ and thus you can use the formula.
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Re: If n = 3^8 - 2^8, which of the following is NOT a factor of  [#permalink]

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$$3^8$$ can be rewritten as $$3^{(4(2))}$$ or $$(3^4 )^2$$. Similarly, $$2^8$$ can be rewritten as $$2^{4(2)}$$ or $$(2^4 )^2$$.

Since both are perfect squares, we can apply the property when working with the difference of two squares:

$$a^2-b^2=(a+b)(a-b)$$

Hence,

$$3^8-2^8=(3^4 )^2-(2^4 )^2$$
$$=(3^4+2^4 )(3^4-2^4 )$$
$$=(81+16)(81-16)$$

We have $$3^8-2^8=97 \times 65= 97 \times 5 \times 13$$. This shows that $$5$$, $$13$$, and $$97$$ are factors of the total.

In addition, since $$7$$ is not a factor of $$3^8-2^8$$, $$35$$ (which is equal to $$5 \times 7$$) can’t be a factor as well.

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

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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

A quick way to solve this it to first recognize that $$3^8 - 2^8$$is a difference of squares, which can be factored.

So, $$n = 3^8 - 2^8$$

$$= (3^4 + 2^4)(3^4 - 2^4)$$

$$= (3^4 + 2^4)(3^2 + 2^2)(3^2 - 2^2)$$

$$= (3^4 + 2^4)(3^2 + 2^2)(3 + 2)(3 - 2)$$

$$= (97)(13)(5)(1)$$

At this point, we can see that 97, 65 (aka 13 x 5), 13 and 5 are all factors of n

Cheers,
Brent
_________________ If n = 3^8 - 2^8, which of the following is NOT a factor of   [#permalink] 06 Mar 2019, 11:06

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