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

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Transcendentalist
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Which of the following integers is NOT a divisor of x if x = (21)(3^7) [#permalink] New post   13 Sep 2013, 09:45
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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 if x = (21)(3^7) [#permalink] New post   20 Feb 2016, 22:00
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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 if x = (21)(3^7) [#permalink] New post   19 Apr 2018, 15:57
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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 if x = (21)(3^7) [#permalink] New post   06 Oct 2014, 16:51
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 if x = (21)(3^7) [#permalink] New post   28 May 2017, 19:34
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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

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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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Re: Which of the following integers is NOT a divisor of x if x = (21)(3^7) [#permalink] New post   24 Nov 2017, 03:14
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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 if x = (21)(3^7) [#permalink] New post   24 Nov 2017, 03:17
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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?

_______________
Yes, that's correct.
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Re: Which of the following integers is NOT a divisor of x if x = (21)(3^7) [#permalink] New post   18 Aug 2018, 08:51
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 if x = (21)(3^7) [#permalink] New post   18 Aug 2018, 09:21
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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 if x = (21)(3^7) [#permalink] New post   10 Apr 2020, 03:59
Expert Reply
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.


We can factor x as:
x = 7 * 3 * 3^7 - 7 * 16 = 7(3^8 - 2^4) = 7(3^4 - 2^2)(3^4 + 2^2) = 7(77)(85) = 7^2 * 11 * 5 * 17

We see that all the numbers in the given answer choices are a factor of x except 15 since 3 is not a factor of x.

Answer: C
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Re: Which of the following integers is NOT a divisor of x if x = (21)(3^7) [#permalink] New post   03 Apr 2021, 07:55
chetan2u 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.


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


Hi chetan2u,

I have a question: how could you disqualify Answer E?
Based on your explanation, I would think that E is wrong as well because the statement is not divisible by 5.

Thank you in advance.
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Re: Which of the following integers is NOT a divisor of x if x = (21)(3^7) [#permalink] New post   03 Apr 2021, 09:48
Expert Reply
argerniki wrote:
chetan2u 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.


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


Hi chetan2u,

I have a question: how could you disqualify Answer E?
Based on your explanation, I would think that E is wrong as well because the statement is not divisible by 5.

Thank you in advance.


Hi

Only two things that we can say from \(21*3^7-112=7*3^8-7*2^4=7(3^8-2^4)\)
1) 7 is surely a divisor.
2) \(3^8-2^4\) tells us that the factor will surely not be 3 or 2.

We cannot say anything about other divisor unless we solve for the term.

Choice C has a 3, so it is surely wrong.
Choice E is 35. Now it does not have factor 3 or 2, so we cannot be sure about it be wrong or right.

Since we know that C is surely wrong, we can take all other choices to be factors.
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Which of the following integers is NOT a divisor of x if x = (21)(3^7) [#permalink] New post   05 Jun 2021, 15:11
Bunuel wrote:
GmatAvengers800 wrote:
Bunuel wrote:

\(21*3^7-112=3^8*7-2^4*7=7(3^8-2^4)=7(3^4-2^2)(3^4+2^2)=7*77*85=7^2*11*5*17\) --> 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?

_______________
Yes, that's correct.


Bunuel - why is the above the case ? How can you say the pink above ?

I ask specifically in relation to the below

if we also consider it as A-B

5 divides A but not B. 2 divides B but not A

Answer could be a multiple of 3

example :A = 10, B = 4. Result of A - B IS a multiple of 3.
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Re: Which of the following integers is NOT a divisor of x if x = (21)(3^7) [#permalink] New post   26 Jun 2021, 07:19
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jabhatta2 wrote:
if we consider it as A-B

3 divides A but not B

so any multiples of 3 cannot divide A-B

why is the above the case ? How can you say the pink above ?


jabhatta2 Consider what this means for the remainder when dividing B by 3: the remainder must be either 1 or 2.
What about the remainder when dividing A by 3? We know for sure the remainder would be 0.
So, the remainder when dividing (A-B) by 3 will be either 2 or 1.
Either way, it's not divisible by 3.
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Re: Which of the following integers is NOT a divisor of x if x = (21)(3^7) [#permalink] New post   28 Jun 2021, 00:54
Breaking down the prime factorization and taking the common factor

X = (7) * [(3)^8 - (2)^4]


Rule: Multiple of n +/- NON-Multiple of n = ALWAYS a NON-Multiple of n


X = 7 * (some integer)


Some integer = Multiple of 3 - Multiple of 2(which is a non multiple of 3)

Thus, some integer will NOT be a multiple of 3 and X therefore will NOT be a multiple of 3


Since 15 = (3) (5)

There is no way that X can be divisible by 15 (since it is not divisible by 3)

Answer 15

Posted from my mobile device
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Re: Which of the following integers is NOT a divisor of x if x = (21)(3^7) [#permalink] New post   16 Jul 2021, 07:58
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Re: Which of the following integers is NOT a divisor of x if x = (21)(3^7) [#permalink]
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