GMAT Question of the Day - Daily to your Mailbox; hard ones only

It is currently 13 Dec 2018, 08:33

R1 Admission Decisions:

Stanford Chat (Calls Started)  |  Wharton Chat  (Calls Expected Soon)  |  Fuqua Chat (Calls Expected Soon)


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
Events & Promotions in December
PrevNext
SuMoTuWeThFrSa
2526272829301
2345678
9101112131415
16171819202122
23242526272829
303112345
Open Detailed Calendar
  • The winning strategy for 700+ on the GMAT

     December 13, 2018

     December 13, 2018

     08:00 AM PST

     09:00 AM PST

    What people who reach the high 700's do differently? We're going to share insights, tips and strategies from data we collected on over 50,000 students who used examPAL.
  • GMATbuster's Weekly GMAT Quant Quiz, Tomorrow, Saturday at 9 AM PST

     December 14, 2018

     December 14, 2018

     09:00 AM PST

     10:00 AM PST

    10 Questions will be posted on the forum and we will post a reply in this Topic with a link to each question. There are prizes for the winners.

If n is a positive integer, is 3^n+3^8 divisible by 4?

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

Hide Tags

Math Revolution GMAT Instructor
User avatar
V
Joined: 16 Aug 2015
Posts: 6629
GMAT 1: 760 Q51 V42
GPA: 3.82
Premium Member
If n is a positive integer, is 3^n+3^8 divisible by 4?  [#permalink]

Show Tags

New post 18 Oct 2017, 02:15
1
1
00:00
A
B
C
D
E

Difficulty:

  75% (hard)

Question Stats:

57% (02:45) correct 43% (02:36) wrong based on 86 sessions

HideShow timer Statistics

[GMAT math practice question]

If n is a positive integer, is \(3^n+3^8\) divisible by 4?

1) n is a multiple of 3.
2) \(3^2 +3^{n+2}\) is divisible by 4

_________________

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only $99 for 3 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"

PS Forum Moderator
avatar
D
Joined: 25 Feb 2013
Posts: 1217
Location: India
GPA: 3.82
GMAT ToolKit User Premium Member Reviews Badge
If n is a positive integer, is 3^n+3^8 divisible by 4?  [#permalink]

Show Tags

New post 18 Oct 2017, 05:28
1
MathRevolution wrote:
[GMAT math practice question]

If n is a positive integer, is \(3^n+3^8\) divisible by 4?

1) n is a multiple of 3.
2) \(3^2 +3^{n+2}\) is divisible by 4


Statement 1: if \(n=3\), then \(3^n+3^8\) = \(3^3+3^8\) => \(3^3(1+3^5)\) => \(3^3*244\), hence divisible by \(4\)

but if \(n=6\), then \(3^n+3^8\) = \(3^6+3^8\) => \(3^6(1+3^2)\) => \(3^6*10\), hence not divisible by \(4\). Hence Insufficient

Statement 2: \(3^2 +3^{n+2}\)= \(3^2(1+3^n)\), now this is a multiple of \(4\), hence can be written as

\(3^2(1+3^n)=4q\), so \(3^n=\frac{4q}{9}-1\), as \(3^n\) is an integer so \(\frac{q}{9}\) must be an integer, so let \(\frac{q}{9}=k\)

Hence \(3^n=4k-1\), substitute this in question stem to get

\(4k-1+3^8 = 4k+(3^4-1)*(3^4+1)=4k+80*82\). Hence divisible by \(4\). Sufficient

Option B
Manager
Manager
avatar
B
Joined: 21 Jun 2014
Posts: 62
Re: If n is a positive integer, is 3^n+3^8 divisible by 4?  [#permalink]

Show Tags

New post 18 Oct 2017, 09:40
The question can be simply narrowed down to " is n odd?" (3 ^n)/4 leaves remainder 3 if n is odd and it leaves remainder 1 of n is even. As 3^8 is leaving remainder 1, we need 3^n to leave remainder 3 and hence n has to be odd.

From statement 1 n can be either odd or even so insuff.

Statement 2 : we can infer that n+2 is odd and hence n is odd sufficient.

OA: B

Sent from my Moto G (5) Plus using GMAT Club Forum mobile app
Math Revolution GMAT Instructor
User avatar
V
Joined: 16 Aug 2015
Posts: 6629
GMAT 1: 760 Q51 V42
GPA: 3.82
Premium Member
Re: If n is a positive integer, is 3^n+3^8 divisible by 4?  [#permalink]

Show Tags

New post 20 Oct 2017, 00:07
=>
Forget conventional ways of solving math questions. In DS, VA (Variable Approach) method is the easiest and quickest way to find the answer without actually solving the problem. Remember that equal number of variables and independent equations ensures a solution.

The first step of VA(Variable Approach) method is modifying the original condition and the question, and rechecking the number of variables and the number of equations.

Since 38 has a remainder 1 when it is divided by 4, the question asks if 3n has a remainder 3 if it is divided by 4.
3^1 = 3 has the remainder 3, when it is divided by 4.
3^2 = 9 has the remainder 1, when it is divided by 4.
3^3 = 27 has the remainder 3, when it is divided by 4.
3^4 = 81 has the remainder 1, when it is divided by 4.
...
Thus, the question asks if n is an odd number.

Condition 1)
Since n is a multiple of 3, it is unknown if n is odd or even.
This is not sufficient.

Condition 2)
“3^2 +3^n+2 is divisible by 4” means 3^n+2 has a remainder of 3 when divided by 4.
It means n + 2 and n are odd integers.
Thus, this condition is sufficient.
Therefore, the answer is B.

Answer: B
_________________

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only $99 for 3 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"

Intern
Intern
avatar
B
Joined: 14 Nov 2012
Posts: 23
GMAT 1: 740 Q51 V38
Reviews Badge
If n is a positive integer, is 3^n+3^8 divisible by 4?  [#permalink]

Show Tags

New post 22 Oct 2017, 11:09
MathRevolution wrote:
[GMAT math practice question]

If n is a positive integer, is \(3^n+3^8\) divisible by 4?

1) n is a multiple of 3.
2) \(3^2 +3^{n+2}\) is divisible by 4



I notice we can apply the Binomial Theorem which is fast and convenient most of the time.
The Binomial Theorem states that, where n is a positive integer:
(a + b)n = an + (nC1)an-1b + (nC2)an-2b2 + … + (nCn-1)abn-1 + bn

With regards to our case :
\(3^n+3^8\) = ( 4 - 1 )^n + (4 - 1) ^ 8 = 4^n + (nC1)*4^(n-1)*(-1)....+ (-1)^n + 4^8 + 8*4^7(-1) + ... + 8*4*(-1)^7 + (-1)^8

As the terms in Italic are divisible by 4, we only have to notice the remaining in Bold: (-1)^n + (-1)^8 = (-1)^n + 1
Case 1: If n is even, then the sum of remainder = 1 + 1 = 2.
Case 2: If n is odd, then the sum of remainder = -1 + 1 = 0

1) n = 3k
If k is even, then n is even.
If k is odd, then n is odd.
Insufficient.

2) \(3^2 +3^{n+2}\) is divisible by 4[/quote]
Again 3^(n+2) +3^2 = [(4 - 1)^(n+2) = 4^(n+2) + ... + (-1)^(n+2)] + (4*2 +1)
The remainder would be (-1)^(n+2) + 1 = 0
=> (-1)^(n+2) = -1
=> (n+2) is odd
=> n is odd => Case 2.
So B is sufficient.
GMAT Club Bot
If n is a positive integer, is 3^n+3^8 divisible by 4? &nbs [#permalink] 22 Oct 2017, 11:09
Display posts from previous: Sort by

If n is a positive integer, is 3^n+3^8 divisible by 4?

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


Copyright

GMAT Club MBA Forum Home| About| Terms and Conditions and Privacy Policy| GMAT Club Rules| Contact| Sitemap

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

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.