Last visit was: 27 Mar 2025, 10:13 It is currently 27 Mar 2025, 10:13
Home
GMAT
Messages
Tests
Schools
Reviews
Rewards
Chat
My Profile
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
Back to Forum
Create Topic
 1   2   
555-605 Level|   Remainders|      
User avatar
LM
User avatar
Joined: 03 Sep 2006
Last visit: 04 Apr 2015
Posts: 445
Own Kudos:
7,330
 [202]
Given Kudos: 33
Posts: 445
Kudos: 7,330
 [202]
If n is a positive integer and r is the remainder when 4 + 7n is divid Updated on: 17 Jun 2019, 04:07
Originally posted by LM on 27 Apr 2010, 09:45.
Last edited by Bunuel on 17 Jun 2019, 04:07, edited 2 times in total.
Edited the question and added the OA
14
Kudos
Add Kudos
187
Bookmarks
Bookmark this Post
00:00
Start Timer
Pause Timer
Resume Timer
Show Answer
Hide
Show
History
A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

25% (medium)

Question Stats:

76% (01:57) correct 24% (02:22) wrong based on 2430 sessions

History

Date
Time
Result
Not Attempted Yet
If n is a positive integer and r is the remainder when 4 + 7n is divided by 3, what is the value of r?

(1) n + 1 is divisible by 3
(2) n > 20.
ShowHide Answer
Official Answer
A
Most Helpful Reply
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 27 March 2025
Posts: 100,113
Own Kudos:
711,349
 [89]
Given Kudos: 92,732
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 100,113
Kudos: 711,349
 [89]
If n is a positive integer and r is the remainder when 4 + 7n is divid 27 Apr 2010, 10:23
22
Kudos
Add Kudos
66
Bookmarks
Bookmark this Post
If n is a positive integer, and r is the remainder when 4 + 7n is divided by 3, what is the value of r?

The condition "r is the remainder when 4 + 7n is divided by 3" can be expressed as \(4 + 7n = 3q + r\), where r is an integer such that \(0 \leq r < 3\). The question asks for the value of r.

(1) n + 1 is divisible by 3

This implies \(n + 1 = 3k\), or \(n = 3k - 1\). Substituting this value of n into the equation, we get \(4 + 7(3k - 1) = 3q + r\). Simplifying, this becomes \(3(7k - 1 - q) = r\). This means that r is a multiple of 3. However, since r is an integer within the range \(0 \leq r < 3\), the only possibility is r = 0. Sufficient.

(2) n > 20.

This is clearly not sufficient. For example, if \(n = 21\), then \(4 + 7n = 151\), resulting in \(r = 1\). However, if \(n = 22\), then \(4 + 7n = 158\), resulting in \(r = 2\).

Answer: A.
User avatar
shrouded1
User avatar
Retired Moderator
Joined: 02 Sep 2010
Last visit: 29 Apr 2018
Posts: 609
Own Kudos:
3,086
 [21]
Given Kudos: 25
Location: London
 Q51  V41
Products:
My Reviews
Posts: 609
Kudos: 3,086
 [21]
If n is a positive integer and r is the remainder when 4 + 7n is divid 30 Sep 2010, 22:37
13
Kudos
Add Kudos
8
Bookmarks
Bookmark this Post
Michmax3
If n is a positive integer and r is the remainder when 4+7n is divided by 3. What is the value of r?

1) n+1 is divisible by 3
2) n>20

7n+4=3(2n+1) + n+1

So remainder when divided by 3 will be same as remainder left by n+1

1) sufficient ... Gives the answer
2) insufficient ... Irrelevant. Eg n= 22,23,24 all are possible

Answer is (a)
General Discussion
avatar
jordanhendrix
avatar
Current Student
Joined: 07 May 2010
Last visit: 19 May 2013
Posts: 722
Own Kudos:
107
 [12]
Given Kudos: 66
Posts: 722
Kudos: 107
 [12]
If n is a positive integer and r is the remainder when 4 + 7n is divid 27 Jul 2010, 16:21
10
Kudos
Add Kudos
2
Bookmarks
Bookmark this Post
i think i may have an easier way....
s1) 7n+4 = (6n+3)+(n+1)
if (n+1)/3 = an integer, so must 3 times (n+1)....which is (6n+3)
s2) Obviously NS
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 27 March 2025
Posts: 100,113
Own Kudos:
711,349
 [18]
Given Kudos: 92,732
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 100,113
Kudos: 711,349
 [18]
If n is a positive integer and r is the remainder when 4 + 7n is divid 27 Jul 2010, 16:40
8
Kudos
Add Kudos
10
Bookmarks
Bookmark this Post
omarjmh
i think i may have an easier way....
s1) 7n+4 = (6n+3)+(n+1)
if (n+1)/3 = an integer, so must 3 times (n+1)....which is (6n+3)
s2) Obviously NS

Little correction: 3 times (n+1) is 3n+3 not (6n+3).

But you are right, we can solve with this approach as well:

(1) n+1 is divisible by 3 --> \(7n+4=(4n+4)+3n=4(n+1)+3n\) --> \(4(n+1)\) is divisible by 3 as \(n+1\) is, and \(3n\) is obviously divisible by 3 as it has 3 as multiple, thus their sum, \(7n+4\), is also divisible by 3, which means that remainder upon division \(7n+4\) by 3 will be 0. Sufficient.

Hope it's clear.
User avatar
Michmax3
User avatar
Joined: 14 Oct 2009
Last visit: 30 Jun 2012
Posts: 353
Own Kudos:
254
Given Kudos: 53
Status:Current Student
Concentration: CPG Marketing
Schools:Chicago Booth 2013, Ross, Duke , Kellogg , Stanford, Haas
 Q41  V42
GPA: 3.8
Products:
My Reviews
Posts: 353
Kudos: 254
If n is a positive integer and r is the remainder when 4 + 7n is divid 30 Sep 2010, 22:45
Kudos
Add Kudos
Bookmarks
Bookmark this Post
shrouded1
Michmax3
7n+4=3(2n+1) + n+1

So remainder when divided by 3 will be same as remainder left by n+1


Can you explain how you get to 7n+4=3(2n+1)?
User avatar
ezhilkumarank
User avatar
Joined: 24 Jun 2010
Last visit: 08 May 2014
Posts: 272
Own Kudos:
714
 [4]
Given Kudos: 50
Status:Time to step up the tempo
Location: Milky way
Concentration: International Business, Marketing
Schools:ISB, Tepper - CMU, Chicago Booth, LSB
Posts: 272
Kudos: 714
 [4]
If n is a positive integer and r is the remainder when 4 + 7n is divid 30 Sep 2010, 22:51
1
Kudos
Add Kudos
3
Bookmarks
Bookmark this Post
The information in the statement A is used favorably to tweak the equation in the question.

Hence 7n+4 becomes 3(2n+1) + (n+1). Now since 3(2n+1) leaves a remainder 0 when divided by 3, the remainder of 7n+4 will be the same as the remainder of (n+1).

Since (n+1) is also given in option A to be divisible by 3, hence remainder 0. Statement A is sufficient.
User avatar
shrouded1
User avatar
Retired Moderator
Joined: 02 Sep 2010
Last visit: 29 Apr 2018
Posts: 609
Own Kudos:
3,086
 [1]
Given Kudos: 25
Location: London
 Q51  V41
Products:
My Reviews
Posts: 609
Kudos: 3,086
 [1]
If n is a positive integer and r is the remainder when 4 + 7n is divid 30 Sep 2010, 22:56
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Michmax3
Can you explain how you get to 7n+4=3(2n+1)?

Just trying to split it out into parts divisible by 3
7n becomes 6n+n
4 becomes 3+1
User avatar
blueseas
User avatar
Current Student
Joined: 14 Dec 2012
Last visit: 15 Jan 2019
Posts: 577
Own Kudos:
4,429
 [7]
Given Kudos: 197
Location: India
Concentration: General Management, Operations
Schools: PGDM (GM) '15 (A)
GMAT 1: 700 Q50 V34
GPA: 3.6
Schools: PGDM (GM) '15 (A)
GMAT 1: 700 Q50 V34
Posts: 577
Kudos: 4,429
 [7]
If n is a positive integer and r is the remainder when 4 + 7n is divid 27 Jul 2013, 12:52
4
Kudos
Add Kudos
3
Bookmarks
Bookmark this Post
forgmat
If n is a positive integer and r is the remainder when 4+7n is divided by 3 what is the value of r ?

1) n+1 is divisible by 3
2) n >20


Please explain

statmnt 1:
n+1 ==>div by 3
therefore 7(n+1) ==>div. by 3
(7n+4)+3 ==>div by ===>this means 7n+4 is div by 3
sufficient

statement 2:
n>20
let n=30==>7n+4=214==>when divided by 3 remainder i s1
let n=40==>7n+4=284==>when divided by 3 remainder i s2
hence not sufficient

hence A
User avatar
TGC
User avatar
Joined: 03 Aug 2012
Last visit: 19 Jul 2017
Posts: 582
Own Kudos:
3,421
 [3]
Given Kudos: 322
Concentration: General Management, General Management
Schools: ISB '16 NUS '17 NTU '17 XLRI SMU '17 IIM Lucknow IPMX'16 (A)
GMAT 1: 630 Q47 V29
GMAT 2: 680 Q50 V32
GPA: 3.7
WE:Information Technology (Finance: Investment Banking)
Schools: ISB '16 NUS '17 NTU '17 XLRI SMU '17 IIM Lucknow IPMX'16 (A)
GMAT 2: 680 Q50 V32
Posts: 582
Kudos: 3,421
 [3]
If n is a positive integer and r is the remainder when 4 + 7n is divid 09 Aug 2013, 09:17
1
Kudos
Add Kudos
2
Bookmarks
Bookmark this Post
Q. What is r?

(1). (n+1) div by 3

By question stem we know that

4 + 7n = 3Q + r

Splitting the equation as below

4+4n+3n = 3Q + r

=> 3n + 4 (n+1) = 3Q + r

LHS is divisible by 3 as (n+1) is div by 3, so RHS should also be divisible by 3 hence r should be 0

(2).

4 + 7n = 3Q +r

Case 1: n=21

4 + 7*21 = 3Q +r

LHS gives 4 as remainder when divided by 3 so r=4

Case 2: n=22

4 + 7*22 = 3Q + r

No info about r, hence inconsistent

(A) it is!
User avatar
vaishnogmat
User avatar
Joined: 10 Jul 2012
Last visit: 27 Oct 2014
Posts: 10
Own Kudos:
271
Given Kudos: 4
Status:Finance Analyst
Affiliations: CPA Australia
Location: Australia
Concentration: Finance, Healthcare
Schools: AGSM '16 (A$)
GMAT 1: 470 Q38 V19
GMAT 2: 600 Q44 V34
GPA: 3.5
WE:Accounting (Healthcare/Pharmaceuticals)
Schools: AGSM '16 (A$)
GMAT 2: 600 Q44 V34
Posts: 10
Kudos: 271
If n is a positive integer and r is the remainder when 4 + 7n is divid 11 Nov 2013, 19:02
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
LM
Please explain the answer......

If n is a positive integer, and r is the remainder when 4 + 7n is divided by 3, what is the value of r?

The condition "r is the remainder when 4 + 7n is divided by 3" can be expressed as \(4 + 7n = 3q + r\), where r is an integer such that \(0 \leq r < 3\). The question asks for the value of r.

(1) n + 1 is divisible by 3

This implies \(n + 1 = 3k\), or \(n = 3k - 1\). Substituting this value of n into the equation, we get \(4 + 7(3k - 1) = 3q + r\). Simplifying, this becomes \(3(7k - 1 - q) = r\). This means that r is a multiple of 3. However, since r is an integer within the range \(0 \leq r < 3\), the only possibility is r = 0. Sufficient.

(2) n > 20.

This is clearly not sufficient. For example, if \(n = 21\), then \(4 + 7n = 151\), resulting in \(r = 1\). However, if \(n = 22\), then \(4 + 7n = 158\), resulting in \(r = 2\).

Answer: A.


I am not sure if it has been asked/discussed before, but can we use the following approach:

re-write 4 + 7n as (3+1) + (6n + 1). now if we divide this by 3 we are left with (1) + (2n +1), which is essentially (2n + 2) ------> 2(n+1)/3 leaves no remainder or in other words 0 - using statement 1 information.

Please correct me if I this approach is incorrect.
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 27 March 2025
Posts: 100,113
Own Kudos:
711,349
 [1]
Given Kudos: 92,732
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 100,113
Kudos: 711,349
 [1]
If n is a positive integer and r is the remainder when 4 + 7n is divid 12 Nov 2013, 02:03
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
vaishnogmat
Bunuel
LM
Please explain the answer......

If n is a positive integer, and r is the remainder when 4 + 7n is divided by 3, what is the value of r?

The condition "r is the remainder when 4 + 7n is divided by 3" can be expressed as \(4 + 7n = 3q + r\), where r is an integer such that \(0 \leq r < 3\). The question asks for the value of r.

(1) n + 1 is divisible by 3

This implies \(n + 1 = 3k\), or \(n = 3k - 1\). Substituting this value of n into the equation, we get \(4 + 7(3k - 1) = 3q + r\). Simplifying, this becomes \(3(7k - 1 - q) = r\). This means that r is a multiple of 3. However, since r is an integer within the range \(0 \leq r < 3\), the only possibility is r = 0. Sufficient.

(2) n > 20.

This is clearly not sufficient. For example, if \(n = 21\), then \(4 + 7n = 151\), resulting in \(r = 1\). However, if \(n = 22\), then \(4 + 7n = 158\), resulting in \(r = 2\).

Answer: A.


I am not sure if it has been asked/discussed before, but can we use the following approach:

re-write 4 + 7n as (3+1) + (6n + 1). now if we divide this by 3 we are left with (1) + (2n +1), which is essentially (2n + 2) ------> 2(n+1)/3 leaves no remainder or in other words 0 - using statement 1 information.

Please correct me if I this approach is incorrect.

4 + 7n = (3+1) + (6n + n) not (3+1) + (6n + 1).

You can solve (1) in another way: \(4 + 7n = 4 + 4n + 3n = 4(n + 1) + 3n\). First statement says that \(n + 1\) is is divisible by 3, thus \(4(n + 1) + 3n = (a \ multiple \ of \ 3) + (a \ multiple \ of \ 3)\). Therefore \(4 + 7n\) yields the remainder of 0, when divided by 3.

Hope it helps.
User avatar
anairamitch1804
User avatar
Joined: 26 Oct 2016
Last visit: 20 Apr 2019
Posts: 506
Own Kudos:
3,487
 [9]
Given Kudos: 877
Location: United States
Concentration: Marketing, International Business
Schools: HBS '19
GMAT 1: 770 Q51 V44
GPA: 4
WE:Education (Education)
Schools: HBS '19
GMAT 1: 770 Q51 V44
Posts: 506
Kudos: 3,487
 [9]
If n is a positive integer and r is the remainder when 4 + 7n is divid 30 Jan 2017, 22:58
6
Kudos
Add Kudos
3
Bookmarks
Bookmark this Post
1) n+1 is a multiple of 3.

If n=2 (we're allowed to pick 2 since 2+1 is a multiple of 3), then (4+14)/3 = 18/3 = 6rem0
If n=5 (we're allowed to pick 5 since 5+1 is a multiple of 3), then (4+35)/3 = 39/3 = 13rem0

at this point you might already be conviced that you'll always get the same answer, but we could try one more just to be safe:

If n=8 (we're allowed to pick 8 since 8+1 is a multiple of 3), then (4+56) = 60/3 = 20rem0

For all 3 plug-ins we get r=0.. sufficient!

2) n > 20

If n=21, then (4+147)/3 = 151/3 = 50rem1
Insuff.

Hence A.
avatar
MariaAntony
avatar
Joined: 10 Mar 2019
Last visit: 14 Sep 2022
Posts: 21
Own Kudos:
4
 [1]
Given Kudos: 296
Location: India
Schools: IMD Jan'23 intake
WE:Project Management (Computer Software)
Schools: IMD Jan'23 intake
Posts: 21
Kudos: 4
 [1]
If n is a positive integer and r is the remainder when 4 + 7n is divid 21 Jul 2021, 08:40
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
Info from the stem:
1. N > 0 and integer
2. 7N + 4 when divided by 3 leaves a remainder R

7N + 4 can be re-written as 6N + 3 + (N + 1). 6N and 3 are divisible by 3 always, so we need to find what is the remainder when N+1 is divided by 3.

S1: N+1 is divisible by 3 - implies R=0 (unique answer). Sufficient.
S2: N > 20 - multiple answers are possible. Insufficient.

Hence the correct answer is option A.
User avatar
avigutman
User avatar
Joined: 17 Jul 2019
Last visit: 03 Oct 2024
Posts: 1,296
Own Kudos:
1,861
 [1]
Given Kudos: 66
Location: Canada
Schools: NYU Stern (WA)
GMAT 1: 780 Q51 V45
GMAT 2: 780 Q50 V47
GMAT 3: 770 Q50 V45
Expert
Expert reply
Schools: NYU Stern (WA)
GMAT 3: 770 Q50 V45
Posts: 1,296
Kudos: 1,861
 [1]
If n is a positive integer and r is the remainder when 4 + 7n is divid 28 Jan 2022, 11:06
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Video solution from Quant Reasoning:
Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1
User avatar
debmishra
User avatar
Joined: 26 Feb 2023
Last visit: 17 Apr 2023
Posts: 1
Given Kudos: 1
Posts: 1
Kudos: 0
If n is a positive integer and r is the remainder when 4 + 7n is divid 14 Mar 2023, 14:10
Kudos
Add Kudos
Bookmarks
Bookmark this Post
I'm thinking of the following approach. Please tell me if it's incorrect.

if r is the reminder when 4+7n is divided by 3 means r = 1(reminder of 4/3) + reminder of 7n/3
Now,
option-1 - if n+1 is divisible by 3, reminder of n when divided by 3 is 2. hence now r = 1 + 2 = 3 which is divisible by 3 and hence r =0. Sufficient.
However,
option-2 - says n > 20. This doesn't mean anything as it can have multiple values which may or may not be divisible by 3. In sufficient.

Answer - A
User avatar
Kimberly77
User avatar
Joined: 16 Nov 2021
Last visit: 07 Sep 2024
Posts: 450
Own Kudos:
41
Given Kudos: 5,901
Location: United Kingdom
GMAT 1: 450 Q42 V34
Products:
My Reviews
GMAT 1: 450 Q42 V34
Posts: 450
Kudos: 41
If n is a positive integer and r is the remainder when 4 + 7n is divid 27 Dec 2023, 15:33
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
If n is a positive integer, and r is the remainder when 4 + 7n is divided by 3, what is the value of r?

The condition "r is the remainder when 4 + 7n is divided by 3" can be expressed as \(4 + 7n = 3q + r\), where r is an integer such that \(0 \leq r < 3\). The question asks for the value of r.

(1) n + 1 is divisible by 3

This implies \(n + 1 = 3k\), or \(n = 3k - 1\). Substituting this value of n into the equation, we get \(4 + 7(3k - 1) = 3q + r\). Simplifying, this becomes \(3(7k - 1 - q) = r\). This means that r is a multiple of 3. However, since r is an integer within the range \(0 \leq r < 3\), the only possibility is r = 0. Sufficient.

(2) n > 20.

This is clearly not sufficient. For example, if \(n = 21\), then \(4 + 7n = 151\), resulting in \(r = 1\). However, if \(n = 22\), then \(4 + 7n = 158\), resulting in \(r = 2\).

Answer: A.

Good explanation Bunuel
One question not sure how you derive below? Could you elaborate please? Thanks

\(4+7(3k-1)=3q+r\) --> \(3(7k-1-q)=r\)
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 27 March 2025
Posts: 100,113
Own Kudos:
711,349
Given Kudos: 92,732
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 100,113
Kudos: 711,349
If n is a positive integer and r is the remainder when 4 + 7n is divid 28 Dec 2023, 00:14
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Kimberly77
Bunuel
If n is a positive integer, and r is the remainder when 4 + 7n is divided by 3, what is the value of r?

The condition "r is the remainder when 4 + 7n is divided by 3" can be expressed as \(4 + 7n = 3q + r\), where r is an integer such that \(0 \leq r < 3\). The question asks for the value of r.

(1) n + 1 is divisible by 3

This implies \(n + 1 = 3k\), or \(n = 3k - 1\). Substituting this value of n into the equation, we get \(4 + 7(3k - 1) = 3q + r\). Simplifying, this becomes \(3(7k - 1 - q) = r\). This means that r is a multiple of 3. However, since r is an integer within the range \(0 \leq r < 3\), the only possibility is r = 0. Sufficient.

(2) n > 20.

This is clearly not sufficient. For example, if \(n = 21\), then \(4 + 7n = 151\), resulting in \(r = 1\). However, if \(n = 22\), then \(4 + 7n = 158\), resulting in \(r = 2\).

Answer: A.

Good explanation Bunuel
One question not sure how you derive below? Could you elaborate please? Thanks

\(4+7(3k-1)=3q+r\) --> \(3(7k-1-q)=r\)

    \(4+7(3k-1)=3q+r\);

    \(4+21k-7=3q+r\);

    \(21k-3=3q+r\);

    \(21k-3-3q=r\);

    \(3(7k-1-q)=r\).

Hope this helps.
User avatar
Kimberly77
User avatar
Joined: 16 Nov 2021
Last visit: 07 Sep 2024
Posts: 450
Own Kudos:
41
Given Kudos: 5,901
Location: United Kingdom
GMAT 1: 450 Q42 V34
Products:
My Reviews
GMAT 1: 450 Q42 V34
Posts: 450
Kudos: 41
If n is a positive integer and r is the remainder when 4 + 7n is divid 28 Dec 2023, 02:40
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
Kimberly77
Bunuel
If n is a positive integer, and r is the remainder when 4 + 7n is divided by 3, what is the value of r?

The condition "r is the remainder when 4 + 7n is divided by 3" can be expressed as \(4 + 7n = 3q + r\), where r is an integer such that \(0 \leq r < 3\). The question asks for the value of r.

(1) n + 1 is divisible by 3

This implies \(n + 1 = 3k\), or \(n = 3k - 1\). Substituting this value of n into the equation, we get \(4 + 7(3k - 1) = 3q + r\). Simplifying, this becomes \(3(7k - 1 - q) = r\). This means that r is a multiple of 3. However, since r is an integer within the range \(0 \leq r < 3\), the only possibility is r = 0. Sufficient.

(2) n > 20.

This is clearly not sufficient. For example, if \(n = 21\), then \(4 + 7n = 151\), resulting in \(r = 1\). However, if \(n = 22\), then \(4 + 7n = 158\), resulting in \(r = 2\).

Answer: A.

Good explanation Bunuel
One question not sure how you derive below? Could you elaborate please? Thanks

\(4+7(3k-1)=3q+r\) --> \(3(7k-1-q)=r\)

    \(4+7(3k-1)=3q+r\);

    \(4+21k-7=3q+r\);

    \(21k-3=3q+r\);

    \(21k-3-3q=r\);

    \(3(7k-1-q)=r\).

Hope this helps.

Ah get it thanks Bunuel :please: , I need to remember to use factoring.
Thanks once again for your great help always !!
avatar
Engineer1
avatar
Joined: 01 Jan 2014
Last visit: 15 Mar 2025
Posts: 210
Own Kudos:
417
Given Kudos: 457
Location: United States (IN)
Concentration: Strategy, Finance
Schools: Booth (Chicago) (A$)
Schools: Booth (Chicago) (A$)
Posts: 210
Kudos: 417
If n is a positive integer and r is the remainder when 4 + 7n is divid 26 Jan 2024, 10:59
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
Kimberly77
Bunuel
If n is a positive integer, and r is the remainder when 4 + 7n is divided by 3, what is the value of r?

The condition "r is the remainder when 4 + 7n is divided by 3" can be expressed as \(4 + 7n = 3q + r\), where r is an integer such that \(0 \leq r < 3\). The question asks for the value of r.

(1) n + 1 is divisible by 3

This implies \(n + 1 = 3k\), or \(n = 3k - 1\). Substituting this value of n into the equation, we get \(4 + 7(3k - 1) = 3q + r\). Simplifying, this becomes \(3(7k - 1 - q) = r\). This means that r is a multiple of 3. However, since r is an integer within the range \(0 \leq r < 3\), the only possibility is r = 0. Sufficient.

(2) n > 20.

This is clearly not sufficient. For example, if \(n = 21\), then \(4 + 7n = 151\), resulting in \(r = 1\). However, if \(n = 22\), then \(4 + 7n = 158\), resulting in \(r = 2\).

Answer: A.

Good explanation Bunuel
One question not sure how you derive below? Could you elaborate please? Thanks

\(4+7(3k-1)=3q+r\) --> \(3(7k-1-q)=r\)

    \(4+7(3k-1)=3q+r\);

    \(4+21k-7=3q+r\);

    \(21k-3=3q+r\);

    \(21k-3-3q=r\);

    \(3(7k-1-q)=r\).

Hope this helps.

Hi Bunuel. Is my method correct? Do we really need to use r in this solution? Thank you!

With statement 1, n+1 = 3K (since n+1 is divisible by 3)
Replacing value of n in the original number 7n+4 = n(3k-1)+4 = 21k - 3 = 3(7k-1), which is divisible by 3. Therefore, remainder is 0.
 1   2   
Moderator:
Bunuel
Math Expert
100113 posts