Summer is Coming! Join the Game of Timers Competition to Win Epic Prizes. Registration is Open. Game starts Mon July 1st.

It is currently 16 Jul 2019, 11:30

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

If z = x^n – 19, is z divisible by 9?

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Find Similar Topics 
Intern
Intern
avatar
Joined: 09 Oct 2012
Posts: 22
Concentration: Strategy
Schools: Bocconi '15 (A)
If z = x^n – 19, is z divisible by 9?  [#permalink]

Show Tags

New post 25 Apr 2013, 01:55
1
1
12
00:00
A
B
C
D
E

Difficulty:

  45% (medium)

Question Stats:

68% (01:46) correct 32% (01:50) wrong based on 302 sessions

HideShow timer Statistics


If z = x^n – 19, is z divisible by 9?

(1) x = 10; n is a positive integer
(2) z + 981 is a multiple of 9
VP
VP
User avatar
Status: Far, far away!
Joined: 02 Sep 2012
Posts: 1044
Location: Italy
Concentration: Finance, Entrepreneurship
GPA: 3.8
GMAT ToolKit User
Re: If z = x^n – 19, is z divisible by 9? 1) x = 10;  [#permalink]

Show Tags

New post 25 Apr 2013, 02:04
1
If \(z = x^n - 19\), is z divisible by 9?

1) x = 10; n is a positive integer
\(z=10^1-19=-9\) divisible by 9
\(z=10^2-19=81\) divisible by 9
\(z=10^3-19=991\) divisible by 9
We can see a pattern here. Whan a number is divisible by nine? When the sum of its digit is a multiple of nine, here is always the case as you can see. Going on with the values of \(n\) you'll see that the resulting number will have the form 9XXX( 9 x times)81 and the sum of those digits will always be a multiple of nine. Sufficient

2) z + 981 is a multiple of 9
this means that \(\frac{z + 981}{9}=\frac{z+9*9*11}{9}=integer\) so one of the factors of z must be 9. Sufficient
D
_________________
It is beyond a doubt that all our knowledge that begins with experience.
Kant , Critique of Pure Reason

Tips and tricks: Inequalities , Mixture | Review: MGMAT workshop
Strategy: SmartGMAT v1.0 | Questions: Verbal challenge SC I-II- CR New SC set out !! , My Quant

Rules for Posting in the Verbal Forum - Rules for Posting in the Quant Forum[/size][/color][/b]
Verbal Forum Moderator
User avatar
B
Joined: 10 Oct 2012
Posts: 605
Re: If z = x^n – 19, is z divisible by 9? 1) x = 10;  [#permalink]

Show Tags

New post 25 Apr 2013, 02:58
rajatr wrote:
If z = x^n – 19, is z divisible by 9?

1) x = 10; n is a positive integer
2) z + 981 is a multiple of 9


From F.S 1, we know that \(z= 10^n -19 = 10^n-10-9 = 10(10^{n-1}-1)-9\) . Thus, z/9 = \(\frac{10(10^{n-1}-1)-9}{(10-1)}\)

\(a^x-1\) is always divisible by (a-1) for positive integer x, thus the above expression is always divisible by 9. Sufficient.

From F.S 2, we know that (z+981) = 9k, where k is an integer. Thus, k = \(\frac{z}{9}+\frac{981}{9}\)=\(\frac{z}{9}\)+ 109. Thus, as k is an integer, thus z/9 has to be an integer,--> z is divisible by 9.Sufficient.

D.
_________________
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 56244
Re: If z = x^n – 19, is z divisible by 9?  [#permalink]

Show Tags

New post 25 Apr 2013, 03:04
2
1
If z = x^n – 19, is z divisible by 9?

(1) x = 10; n is a positive integer --> \(z = x^n - 19=10^n-19=(10^n-1)-18\). Now, 10^n-1 is always a multiple of 9 (for positive integer n, 10^n-1 = 9, 99, 999, ...) and -18 is also a multiple of 9, thus \(x=(10^n-1)-18=(a \ multiple \ of \ 9)-(a \ multiple \ of \ 9)=(a \ multiple \ of \ 9)\). Sufficient.

(2) z + 981 is a multiple of 9. Since 981 is a multiple of 9, then we have that \(z+(a \ multiple \ of \ 9)=(a \ multiple \ of \ 9)\) --> \(z=(a \ multiple \ of \ 9)-(a \ multiple \ of \ 9)=(a \ multiple \ of \ 9)\). Sufficient.

Answer: D.
_________________
CEO
CEO
User avatar
D
Status: GMATINSIGHT Tutor
Joined: 08 Jul 2010
Posts: 2959
Location: India
GMAT: INSIGHT
Schools: Darden '21
WE: Education (Education)
Reviews Badge
If z = x^n - 19, is z divisible by 9?  [#permalink]

Show Tags

New post 14 Jul 2015, 08:27
Priyank38939 wrote:
If z = x^n - 19, is z divisible by 9?
(1) x = 10; n is a positive integer
(2) z + 981 is a multiple of 9


Given : z = x^n - 19

Question : is z divisible by 9?

CONCEPT: The Number will be divisible by 9 if the Sum of the digits of the number is a multiple of 9

Statement 1: x = 10; n is a positive integer

@n=1, z = 10^1 - 19 = -9 Divisible by 9
@n=2, z = 10^2 - 19 = 82 Divisible by 9
@n=3, z = 10^3 - 19 = 981 Divisible by 9
i.e. The result is always Divisible by 9. Hence,
SUFFICIENT

Statement 2: z + 981 is a multiple of 9

981 is a multiple of 9 and also (z + 981) is a multiple of 9 as well which is possible only when z also is a multiple of 9 because if 9a + b = 9c then b = 9(c-1) i.e. a multiple of 9
Hence, z must be a multiple of 9 as well
SUFFICIENT

Answer: Option D
_________________
Prosper!!!
GMATinsight
Bhoopendra Singh and Dr.Sushma Jha
e-mail: info@GMATinsight.com I Call us : +91-9999687183 / 9891333772
Online One-on-One Skype based classes and Classroom Coaching in South and West Delhi
http://www.GMATinsight.com/testimonials.html

ACCESS FREE GMAT TESTS HERE:22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION
Intern
Intern
avatar
Joined: 21 Jun 2016
Posts: 4
Re: If z = x^n – 19, is z divisible by 9?  [#permalink]

Show Tags

New post 05 May 2017, 12:26
Bunuel wrote:
If z = x^n – 19, is z divisible by 9?

(1) x = 10; n is a positive integer --> \(z = x^n - 19=10^n-19=(10^n-1)-18\). Now, 10^n-1 is always a multiple of 9 (for positive integer n, 10^n-1 = 9, 99, 999, ...) and -18 is also a multiple of 9, thus \(x=(10^n-1)-18=(a \ multiple \ of \ 9)-(a \ multiple \ of \ 9)=(a \ multiple \ of \ 9)\). Sufficient.

(2) z + 981 is a multiple of 9. Since 981 is a multiple of 9, then we have that \(z+(a \ multiple \ of \ 9)=(a \ multiple \ of \ 9)\) --> \(z=(a \ multiple \ of \ 9)-(a \ multiple \ of \ 9)=(a \ multiple \ of \ 9)\). Sufficient.

Answer: D.


For statement 2, why can't Z=0 as well as a multiple of 9?
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 56244
Re: If z = x^n – 19, is z divisible by 9?  [#permalink]

Show Tags

New post 05 May 2017, 12:30
1
llamsivel wrote:
Bunuel wrote:
If z = x^n – 19, is z divisible by 9?

(1) x = 10; n is a positive integer --> \(z = x^n - 19=10^n-19=(10^n-1)-18\). Now, 10^n-1 is always a multiple of 9 (for positive integer n, 10^n-1 = 9, 99, 999, ...) and -18 is also a multiple of 9, thus \(x=(10^n-1)-18=(a \ multiple \ of \ 9)-(a \ multiple \ of \ 9)=(a \ multiple \ of \ 9)\). Sufficient.

(2) z + 981 is a multiple of 9. Since 981 is a multiple of 9, then we have that \(z+(a \ multiple \ of \ 9)=(a \ multiple \ of \ 9)\) --> \(z=(a \ multiple \ of \ 9)-(a \ multiple \ of \ 9)=(a \ multiple \ of \ 9)\). Sufficient.

Answer: D.


For statement 2, why can't Z=0 as well as a multiple of 9?


z can be 0 but this won't change the answer because 0 is a multiple of every integer, 0/(non-zero integer) = 0 = integer.
_________________
Manager
Manager
avatar
B
Joined: 03 Sep 2018
Posts: 63
Re: If z = x^n – 19, is z divisible by 9?  [#permalink]

Show Tags

New post 25 Dec 2018, 08:28
I was wondering whether we could also say 19 has a remainder of 1 and 10 has a remainder of 1, hence 10^n-19 has a remainder of 0?
_________________
Please consider giving Kudos if my post contained a helpful reply or question.
GMAT Club Bot
Re: If z = x^n – 19, is z divisible by 9?   [#permalink] 25 Dec 2018, 08:28
Display posts from previous: Sort by

If z = x^n – 19, is z divisible by 9?

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  





Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne