NEW HERE? LEARN MORE
closeclose
Last visit was: 29 Apr 2024, 12:19 It is currently 29 Apr 2024, 12:19
Decision Tracker
My Rewards
New posts
New comers' posts

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.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel

If 3^x - 3^(x - 3) = 2,106, what is the value of x?

SORT BY:
Date
Kudos
Tags:
Find Similar Topics 
Show Tags
Hide Tags
User avatar
B
guddo
Manager
Manager
Joined: 25 May 2021
Posts: 155
Own Kudos [?]: 519 [10]
Given Kudos: 2
Send PM
If 3^x - 3^(x - 3) = 2,106, what is the value of x? [#permalink] New post   26 Jan 2024, 02:18
1
Kudos
9
Bookmarks
00:00
A
B
C
D
E

Difficulty:

35% (medium)

Question Stats:

79% (02:15) correct 21% (02:41) wrong based on 131 sessions
Hide Show timer Statistics
If \(3^x - 3^{x - 3} = 2,106\), what is the value of x?

A. 10
B. 9
C. 8
D. 7
E. 6

Show Spoiler
Attachment:
2024-01-26_13-12-27.png
2024-01-26_13-12-27.png [ 13.13 KiB | Viewed 1877 times ]
ShowHide Answer
Official Answer
D
User avatar
L
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 93014
Own Kudos [?]: 619968 [3]
Given Kudos: 81634
Send PM
CAT Tests Forum Quiz Mode
Re: If 3^x - 3^(x - 3) = 2,106, what is the value of x? [#permalink] New post   26 Jan 2024, 02:23
3
Kudos
Expert Reply
guddo wrote:
If \(3^x - 3^{x - 3} = 2,106\), what is the value of x?

A. 10
B. 9
C. 8
D. 7
E. 6

Show Spoiler
Attachment:
2024-01-26_13-12-27.png


Factor out \(3^{x - 3}\):

    \(3^{x - 3}(3^3 - 1) = 2,106\)

    \(3^{x - 3}26 = 2,106\)

    \(3^{x - 3} = 81\)

    \(3^{x - 3} = 3^4\)

    \(x-3=4\)

    \(x=7\)

Answer: D.
User avatar
S
sachi-in
Manager
Manager
Joined: 12 Oct 2023
Posts: 104
Own Kudos [?]: 48 [1]
Given Kudos: 129
Send PM
Re: If 3^x - 3^(x - 3) = 2,106, what is the value of x? [#permalink] New post   26 Jan 2024, 23:24
1
Bookmarks
This problem is probably related just to calculation ( which will eat up time unless we find a shortcut ) :

3^(x - 3) * (3^3 - 1) = 2106
3^(x - 3) * 26 = 2106
3^(x - 3) * 13 = 1053

We may notice the fact that IFF we multiply 13 with powers of 3 : 3^0 , 3^4 , 3^8 ... it gives 3 in the unit's place
Hence 13 * ( something that is a odd power of 3) 81 fits this nicely since 81*10 810 , hence 81*13 is bound to be 1053 :lol:
User avatar
L
Kinshook
GMAT Club Legend
GMAT Club Legend
Joined: 03 Jun 2019
Posts: 5344
Own Kudos [?]: 3974 [0]
Given Kudos: 160
Location: India
Schools: HBS '22 CBS '22 Wharton '22
GMAT 1: 690 Q50 V34
WE:Engineering (Transportation)
Send PM
Reviews Badge
Re: If 3^x - 3^(x - 3) = 2,106, what is the value of x? [#permalink] New post   26 Jan 2024, 23:42
guddo

Asked: If \(3^x - 3^{x - 3} = 2,106\), what is the value of x?

\(3^{x-3} (3^3 - 1) = 2106 = 2*13*3^4\)
x - 3 = 4
x =7

\(3^7 - 3^4 = 3^4 (3^3 - 1) = 3^4 *26= 2*13*3^4\)

IMO D
User avatar
edthehead7
Intern
Intern
Joined: 13 Feb 2024
Posts: 5
Own Kudos [?]: 0 [0]
Given Kudos: 7
Send PM
Re: If 3^x - 3^(x - 3) = 2,106, what is the value of x? [#permalink] New post   04 Apr 2024, 18:03
­Hi Bunuel - I understand most of the steps but am unclear of the result when factoring. If the original equation is 3^x - 3^x-3 and you are looking to factor out the second segment (3^x-3) .. what happens to the first part? How do we end up with 3^3 - 1 ? ..... thank you
User avatar
L
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 93014
Own Kudos [?]: 619968 [1]
Given Kudos: 81634
Send PM
CAT Tests Forum Quiz Mode
Re: If 3^x - 3^(x - 3) = 2,106, what is the value of x? [#permalink] New post   04 Apr 2024, 21:29
1
Kudos
Expert Reply
 
edthehead7 wrote:
­Hi Bunuel - I understand most of the steps but am unclear of the result when factoring. If the original equation is 3^x - 3^x-3 and you are looking to factor out the second segment (3^x-3) .. what happens to the first part? How do we end up with 3^3 - 1 ? ..... thank you

\(­3^x = 3^{x-3}*3^3\). So, wwhen you factor out \(3^{x-3}\) you are left with only \(3^3\).­
User avatar
P
MartyMurray
Tutor
Joined: 11 Aug 2023
Posts: 837
Own Kudos [?]: 1455 [1]
Given Kudos: 76
GMAT 1: 800 Q51 V51
Send PM
If 3^x - 3^(x - 3) = 2,106, what is the value of x? [#permalink] New post   16 Apr 2024, 10:02
1
Kudos
Expert Reply
If \(3^x - 3^{x - 3} = 2,106\), what is the value of x?

Since the maximum value of x is 10, which is relatively low, we can solve this problem using a units digits pattern.

The units digit pattern for powers of 3 is the following:

3 9 7 1 3 9 7 1 3 9

So, \(3^{10}\) has a units digit of 9.

Since 2,106 has a units digit of 6, we can now check to see which units digits are 3 apart that have a difference of 6.

9 - 7 = not 6

3 - 9 = not 6

1 - 3 = not 6

7 - 1 = 6

So, counting the units digits from the left to the second 7, we find that x = 7

A. 10
B. 9
C. 8
D. 7
E. 6­

Correct answer: D­
GMAT Club Bot
gmatclubot
If 3^x - 3^(x - 3) = 2,106, what is the value of x? [#permalink]
16 Apr 2024, 10:02
Moderators:
Bunuel
Math Expert
93015 posts
yashikaaggarwal
Senior Moderator - Masters Forum
3137 posts

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne
Main navigation
GMAT Resources
Partners
MBA Resources
www.gmac.com|www.mba.com | | GMAT Club Rules | Terms and Conditions