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Number Property Question

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by nbalow » Wed Mar 25, 2015 3:11 pm
Could one of the experts give me a quick way to get through a problem such as the following:

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

Please walk me through how you would attack this.

Thank you
Nick
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by Brent@GMATPrepNow » Wed Mar 25, 2015 3:41 pm
nbalow wrote:Could one of the experts give me a quick way to get through a problem such as the following:

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
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
Brent Hanneson - Creator of GMATPrepNow.com
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