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

 It is currently 19 Sep 2019, 16:58 ### 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

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. ### Request Expert Reply # A group of n students can be divided into equal groups of 4

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

### Hide Tags

Senior Manager  Status: Finally Done. Admitted in Kellogg for 2015 intake
Joined: 25 Jun 2011
Posts: 458
Location: United Kingdom
Concentration: International Business, Strategy
GMAT 1: 730 Q49 V45 GPA: 2.9
WE: Information Technology (Consulting)
A group of n students can be divided into equal groups of 4  [#permalink]

### Show Tags

12 00:00

Difficulty:   45% (medium)

Question Stats: 68% (02:09) correct 32% (02:15) wrong based on 434 sessions

### HideShow timer Statistics

A group of n students can be divided into equal groups of 4 with 1 student left over or equal groups of 5 with 3 students left over. What is the sum of the two smallest possible values of n?

A. 33
B. 46
C. 49
D. 53
E. 86
OA is a and this is how I arrived at.

Let say n = 4x+1 and n = 5y+3 -----> From the question Stem

n=4x+1 n = 5y+3
5 8
9 13
13 18
17 23
25 33
29 38
33 43

I get these above values by putting the same values for x and y. Is my concept correct?

_________________
Best Regards,
E.

MGMAT 1 --> 530
MGMAT 2--> 640
MGMAT 3 ---> 610
GMAT ==> 730

Originally posted by enigma123 on 21 Jan 2012, 14:37.
Last edited by mau5 on 18 Nov 2013, 22:35, edited 3 times in total.
Edited the OA
Math Expert V
Joined: 02 Sep 2009
Posts: 58117
Re: Group of Students  [#permalink]

### Show Tags

4
enigma123 wrote:
A group of n students can be divided into equal groups of 4 with 1 student left over or equal groups of 5 with 3 students left over. What is the sum of the two smallest possible values of n?
a) 33
b) 46
c) 49
d) 53
e) 86

Yes you can do the way you started by listing the possible values of n for both patterns and then picking first two matching numbers from these lists. Since we are dealing with easy and small numbers this approach probably would be the fastest one.

A group of n students can be divided into equal groups of 4 with 1 student left over --> n=4q+1 --> n can be: 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, ... (basically an evenly spaced set with common difference of 4)

A group of n students can be divided into equal groups of 5 with 3 students left over --> n=5p+3 --> n can be: 3, 8, 13, 18, 23, 28, 33, 38, ... (basically an evenly spaced set with common difference of 5)

Therefor two smallest possible values of n are 13 and 33 --> 13+33=46.

Else you can derive general formula based on n=4q+1 and n=5p+3.

Divisor will be the least common multiple of above two divisors 4 and 5, hence 20.

Remainder will be the first common integer in above two patterns, hence 13 --> so, to satisfy both conditions, n must be of a type n=20m+13: 13, 33, 53, ... (two two smallest possible values of n are for m=0 and for m=1, so 13, and 33 respectively) --> 13+33=46.

manhattan-remainder-problem-93752.html#p721341
when-positive-integer-n-is-divided-by-5-the-remainder-is-90442.html#p722552
when-the-positive-integer-a-is-divided-by-5-and-125591.html#p1028654

Hope it helps.
_________________
Senior Manager  Status: Finally Done. Admitted in Kellogg for 2015 intake
Joined: 25 Jun 2011
Posts: 458
Location: United Kingdom
Concentration: International Business, Strategy
GMAT 1: 730 Q49 V45 GPA: 2.9
WE: Information Technology (Consulting)
Re: A group of n students can be divided into equal groups of 4  [#permalink]

### Show Tags

Hi Bunuel - can the values of q and p be ZERO? I don't think they can be and therefore n cannot be 1 & 3. Am I wrong?
_________________
Best Regards,
E.

MGMAT 1 --> 530
MGMAT 2--> 640
MGMAT 3 ---> 610
GMAT ==> 730
Math Expert V
Joined: 02 Sep 2009
Posts: 58117
Re: A group of n students can be divided into equal groups of 4  [#permalink]

### Show Tags

2
enigma123 wrote:
Hi Bunuel - can the values of q and p be ZERO? I don't think they can be and therefore n cannot be 1 & 3. Am I wrong?

THEORY:
Positive integer $$a$$ divided by positive integer $$d$$ yields a reminder of $$r$$ can always be expressed as $$a=qd+r$$, where $$q$$ is called a quotient and $$r$$ is called a remainder, note here that $$0\leq{r}<d$$ (remainder is non-negative integer and always less than divisor).

For example we are told that when positive integer n is divided by 25, the remainder is 13 --> $$n=25q+13$$. Now, the lowest value of $$q$$ can be zero and in this case $$n=13$$ --> 13 divided by 25 yields the remainder of 13. Generally when divisor (25 in our case) is more than dividend (13 in our case) then the reminder equals to the dividend. For example:
3 divided by 24 yields a reminder of 3 --> $$3=0*24+3$$;
or:
5 divided by 6 yields a reminder of 5 --> $$5=0*6+5$$.

Also note that you shouldn't worry about negative numbers in divisibility questions, as every GMAT divisibility question will tell you in advance that any unknowns represent positive integers.

OUR ORIGINAL QUESTION:
We are told that "a group of n students can be divided into equal groups of 4 with 1 student left over" --> n=4q+1. Here q also can be zero, which would mean that there is only 1 student and zero groups of 4.

QUESTIONS TO PRACTICE:
PS questions on remainders: search.php?search_id=tag&tag_id=199
DS questions on remainders: search.php?search_id=tag&tag_id=198

THEORY ON REMAINDERS: compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html

Hope it helps.
_________________
Senior Manager  Status: Finally Done. Admitted in Kellogg for 2015 intake
Joined: 25 Jun 2011
Posts: 458
Location: United Kingdom
Concentration: International Business, Strategy
GMAT 1: 730 Q49 V45 GPA: 2.9
WE: Information Technology (Consulting)
Re: A group of n students can be divided into equal groups of 4  [#permalink]

### Show Tags

You are a true genius buddy.
_________________
Best Regards,
E.

MGMAT 1 --> 530
MGMAT 2--> 640
MGMAT 3 ---> 610
GMAT ==> 730
Manager  Joined: 12 Nov 2011
Posts: 59
Re: A group of n students can be divided into equal groups of 4  [#permalink]

### Show Tags

Stright and simple B
just try numbers for 4*n+1=5*k+3 where k and n are integers, find 2 smallest and your solve it for 30 sec
Veritas Prep GMAT Instructor D
Joined: 16 Oct 2010
Posts: 9643
Location: Pune, India
Re: A group of n students can be divided into equal groups of 4  [#permalink]

### Show Tags

2
enigma123 wrote:
A group of n students can be divided into equal groups of 4 with 1 student left over or equal groups of 5 with 3 students left over. What is the sum of the two smallest possible values of n?
a)33
b)46
c)49
d)53
e) 86

OA is a and this is how I arrived at.

Let say n = 4x+1 and n = 5y+3 -----> From the question Stem

n=4x+1 n = 5y+3
5 8
9 13
13 18
17 23
25 33
29 38
33 43

I get these above values by putting the same values for x and y. Is my concept correct?

I wrote a blog post discussing this concept in detail. I have discussed a couple of questions very similar to this one in the post. You can check it out if you like.

http://www.veritasprep.com/blog/2011/05 ... s-part-ii/
_________________
Karishma
Veritas Prep GMAT Instructor

Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >
Manager  Status: D-Day is on February 10th. and I am not stressed
Affiliations: American Management association, American Association of financial accountants
Joined: 12 Apr 2011
Posts: 162
Location: Kuwait
Schools: Columbia university
Re: A group of n students can be divided into equal groups of 4  [#permalink]

### Show Tags

1
n=4q+1 -----> 1,5,9,13,17,21,25,29,33

n=5q+3------->3,8,13,18,23,28,33

the first two common numbers are 13 and 33, so add those numbers, you get 13+33=46

so, asnwer is B, 46

hope this helps
_________________
Sky is the limit
Senior Manager  Status: Finally Done. Admitted in Kellogg for 2015 intake
Joined: 25 Jun 2011
Posts: 458
Location: United Kingdom
Concentration: International Business, Strategy
GMAT 1: 730 Q49 V45 GPA: 2.9
WE: Information Technology (Consulting)
Re: A group of n students can be divided into equal groups of 4  [#permalink]

### Show Tags

Thanks Karishma. A very helpful post.
_________________
Best Regards,
E.

MGMAT 1 --> 530
MGMAT 2--> 640
MGMAT 3 ---> 610
GMAT ==> 730
Manager  Joined: 28 Apr 2013
Posts: 120
Location: India
GPA: 4
WE: Medicine and Health (Health Care)
Re: Group of Students  [#permalink]

### Show Tags  Again since the max student left over are 3 so any no with 3 can be common between the two groups ; 3, 13, 23, 33, 43, 53 etc.
using in the formula pf n= 4q+1 = 5r+ 3 ; will get 13 and 33 ; addd then =46
_________________
Thanks for Posting

LEARN TO ANALYSE

+1 kudos if you like
Manager  Joined: 26 Sep 2013
Posts: 188
Concentration: Finance, Economics
GMAT 1: 670 Q39 V41 GMAT 2: 730 Q49 V41 Re: Group of Students  [#permalink]

### Show Tags

Bunuel wrote:
enigma123 wrote:
A group of n students can be divided into equal groups of 4 with 1 student left over or equal groups of 5 with 3 students left over. What is the sum of the two smallest possible values of n?
a) 33
b) 46
c) 49
d) 53
e) 86

Yes you can do the way you started by listing the possible values of n for both patterns and then picking first two matching numbers from these lists. Since we are dealing with easy and small numbers this approach probably would be the fastest one.

A group of n students can be divided into equal groups of 4 with 1 student left over --> n=4q+1 --> n can be: 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, ... (basically an evenly spaced set with common difference of 4)

A group of n students can be divided into equal groups of 5 with 3 students left over --> n=5p+3 --> n can be: 3, 8, 13, 18, 23, 28, 33, 38, ... (basically an evenly spaced set with common difference of 5)

Therefor two smallest possible values of n are 13 and 33 --> 13+33=46.

Else you can derive general formula based on n=4q+1 and n=5p+3.

Divisor will be the least common multiple of above two divisors 4 and 5, hence 20.

Remainder will be the first common integer in above two patterns, hence 13 --> so, to satisfy both conditions, n must be of a type n=20m+13: 13, 33, 53, ... (two two smallest possible values of n are for m=0 and for m=1, so 13, and 33 respectively) --> 13+33=46.

manhattan-remainder-problem-93752.html#p721341
when-positive-integer-n-is-divided-by-5-the-remainder-is-90442.html#p722552
when-the-positive-integer-a-is-divided-by-5-and-125591.html#p1028654

Hope it helps.

Could you explain what you mean by 'divisor'? Where is there a divisor in those two equations....Is there work here that you did in your head, but left out? I'm trying to learn how to do these, so if you could show any work that was done mentally I would appreciate it! Thanks!
Math Expert V
Joined: 02 Sep 2009
Posts: 58117
Re: Group of Students  [#permalink]

### Show Tags

AccipiterQ wrote:
Bunuel wrote:
enigma123 wrote:
A group of n students can be divided into equal groups of 4 with 1 student left over or equal groups of 5 with 3 students left over. What is the sum of the two smallest possible values of n?
a) 33
b) 46
c) 49
d) 53
e) 86

Yes you can do the way you started by listing the possible values of n for both patterns and then picking first two matching numbers from these lists. Since we are dealing with easy and small numbers this approach probably would be the fastest one.

A group of n students can be divided into equal groups of 4 with 1 student left over --> n=4q+1 --> n can be: 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, ... (basically an evenly spaced set with common difference of 4)

A group of n students can be divided into equal groups of 5 with 3 students left over --> n=5p+3 --> n can be: 3, 8, 13, 18, 23, 28, 33, 38, ... (basically an evenly spaced set with common difference of 5)

Therefor two smallest possible values of n are 13 and 33 --> 13+33=46.

Else you can derive general formula based on n=4q+1 and n=5p+3.

Divisor will be the least common multiple of above two divisors 4 and 5, hence 20.

Remainder will be the first common integer in above two patterns, hence 13 --> so, to satisfy both conditions, n must be of a type n=20m+13: 13, 33, 53, ... (two two smallest possible values of n are for m=0 and for m=1, so 13, and 33 respectively) --> 13+33=46.

manhattan-remainder-problem-93752.html#p721341
when-positive-integer-n-is-divided-by-5-the-remainder-is-90442.html#p722552
when-the-positive-integer-a-is-divided-by-5-and-125591.html#p1028654

Hope it helps.

Could you explain what you mean by 'divisor'? Where is there a divisor in those two equations....Is there work here that you did in your head, but left out? I'm trying to learn how to do these, so if you could show any work that was done mentally I would appreciate it! Thanks!

20 is the divisor in n=20m+13. Please follow the links in my post.
_________________
SVP  P
Joined: 12 Dec 2016
Posts: 1502
Location: United States
GMAT 1: 700 Q49 V33 GPA: 3.64
Re: A group of n students can be divided into equal groups of 4  [#permalink]

### Show Tags

try every combination
VP  P
Joined: 07 Dec 2014
Posts: 1233
Re: A group of n students can be divided into equal groups of 4  [#permalink]

### Show Tags

1
enigma123 wrote:
A group of n students can be divided into equal groups of 4 with 1 student left over or equal groups of 5 with 3 students left over. What is the sum of the two smallest possible values of n?

A. 33
B. 46
C. 49
D. 53
E. 86

assume difference of 1 between quotients
(n-1)/4-(n-3)/5=1
n=13=smallest n
13+4*5=33=second smallest n
13+33=46
B
SVP  P
Joined: 12 Dec 2016
Posts: 1502
Location: United States
GMAT 1: 700 Q49 V33 GPA: 3.64
Re: A group of n students can be divided into equal groups of 4  [#permalink]

### Show Tags

gracie wrote:
enigma123 wrote:
A group of n students can be divided into equal groups of 4 with 1 student left over or equal groups of 5 with 3 students left over. What is the sum of the two smallest possible values of n?

A. 33
B. 46
C. 49
D. 53
E. 86

assume difference of 1 between quotients
(n-1)/4-(n-3)/5=1
n=13=smallest n
13+4*5=33=second smallest n
13+33=46
B

this can be applied to any problem of this kind?
SVP  P
Joined: 12 Dec 2016
Posts: 1502
Location: United States
GMAT 1: 700 Q49 V33 GPA: 3.64
Re: A group of n students can be divided into equal groups of 4  [#permalink]

### Show Tags

gracie wrote:
enigma123 wrote:
A group of n students can be divided into equal groups of 4 with 1 student left over or equal groups of 5 with 3 students left over. What is the sum of the two smallest possible values of n?

A. 33
B. 46
C. 49
D. 53
E. 86

assume difference of 1 between quotients
(n-1)/4-(n-3)/5=1
n=13=smallest n
13+4*5=33=second smallest n
13+33=46
B

also, how can you assume that the difference is 1, it can be 2 or 3 or 4
Director  G
Joined: 19 Oct 2013
Posts: 524
Location: Kuwait
GPA: 3.2
WE: Engineering (Real Estate)
Re: A group of n students can be divided into equal groups of 4  [#permalink]

### Show Tags

$$\frac{n}{x}= 4 + \frac{1}{x}$$

$$n = 4x + 1$$

n = 5,9,13,17,21,25,29,33

$$\frac{n}{y}= 5 + \frac{3}{y}$$

$$n = 5y + 3$$

n = 8,13,18,23,28,33

13 + 33 = 46

Intern  B
Joined: 09 Apr 2018
Posts: 31
GPA: 4
A group of n students can be divided into equal groups of 4  [#permalink]

### Show Tags

Deriving the general formula n=kx+r, based on both statements, where k is the LCM of the 2 divisors and r is the first common integer:

n=4q+1-> n=1,5,9,13...
n=5p+3-> n=3,8,13...

general formula: n=20x+13-> n=13, 33...

sum=13+33=46

Please hit Kudos if you liked this approach  A group of n students can be divided into equal groups of 4   [#permalink] 14 Oct 2018, 23:16
Display posts from previous: Sort by

# A group of n students can be divided into equal groups of 4

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

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