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

Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pm; Location: Vancouver, BC; Thanked: 5254 times; Followed by:1268 members; GMAT Score:770

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

by Brent@GMATPrepNow » Thu Apr 02, 2020 6:59 am

BTGModeratorVI wrote: ↑
Tue Mar 31, 2020 5:09 am
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

Answer: C
Source: Manhattan prep

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

Quote

GMAT/MBA Expert

Scott@TargetTestPrep: GMAT Instructor; Posts: 8088; Joined: Sat Apr 25, 2015 10:56 am; Location: Los Angeles, CA; Thanked: 43 times; Followed by:29 members

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

by Scott@TargetTestPrep » Mon Apr 13, 2020 4:01 am

BTGModeratorVI wrote: ↑
Tue Mar 31, 2020 5:09 am
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

Answer: C
Source: Manhattan prep

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

Scott Woodbury-Stewart
Founder and CEO
[email protected]