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

Question

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

chetan2u · Accepted Answer

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

BrentGMATPrepNow · Answer

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

intheend14 · Answer

Thanks Bunuel! More elegant and less time-consuming than my answer!

mbaapp1234 · Answer

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.

GmatAvengers800 · Answer

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?

Bunuel · Answer

_______________
Yes, that's correct.

mohitdongaonkar · Answer

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 ?

BrentGMATPrepNow · Answer

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

ScottTargetTestPrep · Answer

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

argerniki · Answer

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.

chetan2u · Answer

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.

jabhatta2 · Answer

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.

avigutman · Answer

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.

Fdambro294 · Answer

Breaking down the prime factorization and taking the common factor

X = (7) *

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

avigutman · Answer

Video solution from Quant Reasoning starts at 17:55 Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1 https://www.youtube.com/watch?v=cx-mlMq7TGI

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

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

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