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Which of the following integers is NOT a divisor of x

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Which of the following integers is NOT a divisor of x  [#permalink]

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New post 13 Sep 2013, 08:45
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A
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Question Stats:

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Which of the following integers is NOT a divisor of x if x = (21)(3^7) – (112)?

A 7
B 11
C 15
D 17
E 35

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Re: Which of the following integers is NOT a divisor of x  [#permalink]

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New post 13 Sep 2013, 08:52
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3
Transcendentalist wrote:
Which of the following integers is NOT a divisor of x if x = (21)(3^7) – (112)?

A 7
B 11
C 15
D 17
E 35

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\(21*3^7-112=3^8*7-2^4*7=7(3^8-2^4)=7(3^4-2^2)(3^4+2^2)=7*77*85=7^2*11*5*17\) --> the product is NOT divisible by 3, thus it's not divisible by 15=3*5 either.

Answer: C.
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Re: Which of the following integers is NOT a divisor of x  [#permalink]

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New post 06 Oct 2014, 15:51
Thanks Bunuel! More elegant and less time-consuming than my answer!
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Re: Which of the following integers is NOT a divisor of x  [#permalink]

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New post 20 Feb 2016, 11:41
Does anyone have some similar practice problems? Thanks a lot!
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Re: Which of the following integers is NOT a divisor of x  [#permalink]

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New post 20 Feb 2016, 21:00
4
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Transcendentalist wrote:
Which of the following integers is NOT a divisor of x if x = (21)(3^7) – (112)?

A 7
B 11
C 15
D 17
E 35

Press Kudos if you like the question.


Hi,
here, we donot require to do any calculations at all...
the choices itself give away the answer...


merely playing with choices given, this 700 level Q can turn into a SUB-600 level Q..
In (21)(3^7) – (112).. (21)(3^7) is a multiple of both 3 and 7, but 112 is a multiple of 7 and not 3..
so addition or subtraction of these terms will be multiple of 7, but not of 3..

However C is 15, which is a multiple of 3,.. hence not possible

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Re: Which of the following integers is NOT a divisor of x  [#permalink]

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New post 22 Feb 2016, 00:19
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Re: Which of the following integers is NOT a divisor of x  [#permalink]

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New post 22 Feb 2016, 01:48
Bunuel wrote:
lpetroski wrote:
Does anyone have some similar practice problems? Thanks a lot!


Check our question bank: viewforumtags.php

Also, check similar questions block at the end of this page.

Hope it helps.



Hi Brunel - actually I tried both those options before even posting this request with no success! I was hoping for some problems that are exactly like this one, but if you don't have any easily accessible, no problem. Since I can't find them I assume this is not a usual problem that would come up. Thanks anyway.
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Re: Which of the following integers is NOT a divisor of x  [#permalink]

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New post 28 May 2017, 18:34
1
Transcendentalist wrote:
Which of the following integers is NOT a divisor of x if x = (21)(3^7) – (112)?

A 7
B 11
C 15
D 17
E 35

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First break the term down into its prime factors

x = 21*3^7 -112 = 3*7*3^7 -112 = 3^8*7 -112

x = 3^8*7-7*16

x = 7(3^8 - 16)

Now it's important to realize that 3^8 - 16 is a perfect square, albeit a hidden one. You can eliminate answer choice A.

x = 7(3^4-4)(3^4+4). Once again, notice that 3^4-4 is a perfect square.

x = 7(3^2+2)(3^2-2)(3^4+4).

At this stage, you can start to convert into the non-prime values

x = 7(11)(7)(85)

85 = 17*5, so only 15 is not a divisor of x.
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Which of the following integers is NOT a divisor of x  [#permalink]

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New post 24 Nov 2017, 02:14
3
Bunuel wrote:
Transcendentalist wrote:
Which of the following integers is NOT a divisor of x if x = (21)(3^7) – (112)?

A 7
B 11
C 15
D 17
E 35

Press Kudos if you like the question.


\(21*3^7-112=3^8*7-2^4*7=7(3^8-2^4)=7(3^4-2^2)(3^4+2^2)=7*77*85=7^2*11*5*17\) --> the product is NOT divisible by 3, thus it's not divisible by 15=3*5 either.

Answer: C.


Hi Bunuel

Another way would be

if we consider it as A-B

3 divides A but not B

so any multiples of 3 cannot divide A-B

am I correct in my approach?
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Re: Which of the following integers is NOT a divisor of x  [#permalink]

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New post 24 Nov 2017, 02:17
1
1
GmatAvengers800 wrote:
Bunuel wrote:
Transcendentalist wrote:
Which of the following integers is NOT a divisor of x if x = (21)(3^7) – (112)?

A 7
B 11
C 15
D 17
E 35

Press Kudos if you like the question.


\(21*3^7-112=3^8*7-2^4*7=7(3^8-2^4)=7(3^4-2^2)(3^4+2^2)=7*77*85=7^2*11*5*17\) --> the product is NOT divisible by 3, thus it's not divisible by 15=3*5 either.

Answer: C.


Hi Bunuel

Another way would be

if we consider it as A-B

3 divides A but not B

so any multiples of 3 cannot divide A-B

am I correct in my approach?

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Yes, that's correct.
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Re: Which of the following integers is NOT a divisor of x  [#permalink]

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New post 19 Apr 2018, 14:57
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Transcendentalist wrote:
Which of the following integers is NOT a divisor of x if x = (21)(3^7) – (112)?

A 7
B 11
C 15
D 17
E 35

Press Kudos if you like the question.


Let's factor this thing like crazy.
(21)(3^7) – (112) = (3)(7)(3^7) – (7)(2)(2)(2)(2)
= (7)(3^8) – (7)(2^4)
= 7(3^8 - 2^4) NOTE: The part in the brackets is a difference of squares, since 3^8 = (3^4)^2 and 2^4 = (2^2)^2
= 7(3^4 + 2^2)(3^4 - 2^2) we factored the difference of squares
= 7(3^4 + 2^2)(3^2 - 2)(3^2 + 2) we factored another difference of squares
= 7(81 + 4)(9 - 2)(9 + 2) evaluated
= 7(85)(7)(11) evaluated
= (7)(5)(17)(7)(11)

We can see that the above product is divisible by 7, 11, 17, and 35
BUT it is not divisible by 15

Answer: C

Cheers,
Brent
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Re: Which of the following integers is NOT a divisor of x  [#permalink]

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New post 18 Aug 2018, 07:51
GMATPrepNow wrote:
Transcendentalist wrote:
Which of the following integers is NOT a divisor of x if x = (21)(3^7) – (112)?

A 7
B 11
C 15
D 17
E 35

Press Kudos if you like the question.


Let's factor this thing like crazy.
(21)(3^7) – (112) = (3)(7)(3^7) – (7)(2)(2)(2)(2)
= (7)(3^8) – (7)(2^4)
= 7(3^8 - 2^4) NOTE: The part in the brackets is a difference of squares, since 3^8 = (3^4)^2 and 2^4 = (2^2)^2
= 7(3^4 + 2^2)(3^4 - 2^2) we factored the difference of squares
= 7(3^4 + 2^2)(3^2 - 2)(3^2 + 2) we factored another difference of squares
= 7(81 + 4)(9 - 2)(9 + 2) evaluated
= 7(85)(7)(11) evaluated
= (7)(5)(17)(7)(11)

We can see that the above product is divisible by 7, 11, 17, and 35
BUT it is not divisible by 15

Answer: C

Cheers,
Brent



Hi Brent,

slightly off topic question but pls answer if you can.

11 or 17 are not divisors of 21*3^7 or 112.
here k (11 or 17) is not a factor of M (21*3^7) or N (112).
then how can k be factor of x which is M-N ?
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Re: Which of the following integers is NOT a divisor of x  [#permalink]

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New post 18 Aug 2018, 08:21
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mohitdongaonkar wrote:
Hi Brent,

slightly off topic question but pls answer if you can.

11 or 17 are not divisors of 21*3^7 or 112.
here k (11 or 17) is not a factor of M (21*3^7) or N (112).
then how can k be factor of x which is M-N ?


I think you have combined two rules to make a new (but incorrect) rule.
Rule: If d is a factor of j, and d is a factor of k, then d is a factor of j-k (as well as a factor of j+k)
Rule: If d is a factor of j, but d is NOT a factor of k, then d is NOT a factor of j-k

There is no rule that says:
If d is NOT a factor of j, and d is NOT a factor of k, then d is NOT a factor of j-k
Here's a counter-example:
3 is not a factor of 22, 3 is not a factor of 7. HOWEVER, 3 IS a factor of 22-7 (aka 15)

Does that help?

Cheers,
Brent
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Re: Which of the following integers is NOT a divisor of x  [#permalink]

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New post 18 Aug 2018, 09:20
Transcendentalist wrote:
Which of the following integers is NOT a divisor of x if x = (21)(3^7) – (112)?

A 7
B 11
C 15
D 17
E 35

Press Kudos if you like the question.


\(OA:C\)
\(x = (21)(3^7) – (112)\)
\(x = 7[(3)(3^7) – (16)] = 7(3^8-4^2)=7(3^4+4)(3^4-4) =7(81+4)(81-4)=7*85*77=5^1*7^2*11^1*17^1\)
\(15\) is only option which cannot be divisor of \((21)(3^7) – (112)\) as there is no \(3\) in prime of factorisation of it
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Re: Which of the following integers is NOT a divisor of x  [#permalink]

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New post 16 Nov 2018, 03:50
There is a shortcut to solve this question. The integer 21*3^7 is divisible by 3 and 112 is not divisible by 3. If we subtract or add any multiple of 3 by a non multiple of 3, the result will always be a non multiple of 3.

Example - 9 - 2. This results in 7. 9 is a multiple of 3 while 2 is not.

In the answer choices only one option is divisible by 3 and that is 15. Option C. Hence C is the answer.
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Re: Which of the following integers is NOT a divisor of x &nbs [#permalink] 16 Nov 2018, 03:50
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