GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

 It is currently 11 Jul 2020, 14:00

### 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

# Let n and k be positive integers with k ≤ n. From an n × n array of do

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

### Hide Tags

Senior Manager
Joined: 04 Sep 2017
Posts: 318
Let n and k be positive integers with k ≤ n. From an n × n array of do  [#permalink]

### Show Tags

21 Sep 2019, 03:30
1
53
00:00

Difficulty:

95% (hard)

Question Stats:

37% (03:04) correct 63% (02:56) wrong based on 468 sessions

### HideShow timer Statistics

Let n and k be positive integers with k ≤ n. From an n × n array of dots, a k × k array of dots is selected. The figure above shows two examples where the selected k × k array is enclosed in a square. How many pairs (n, k) are possible so that exactly 48 of the dots in the n × n array are NOT in the selected k × k array?

A. 1
B. 2
C. 3
D. 4
E. 5

PS03551.01

Attachment:

2019-09-21_1528.png [ 15.95 KiB | Viewed 8429 times ]
##### Most Helpful Expert Reply
Math Expert
Joined: 02 Aug 2009
Posts: 8757
Let n and k be positive integers with k ≤ n. From an n × n array of do  [#permalink]

### Show Tags

19 Oct 2019, 05:17
11
1
10
gmatt1476 wrote:

Let n and k be positive integers with k ≤ n. From an n × n array of dots, a k × k array of dots is selected. The figure above shows two examples where the selected k × k array is enclosed in a square. How many pairs (n, k) are possible so that exactly 48 of the dots in the n × n array are NOT in the selected k × k array?

A. 1
B. 2
C. 3
D. 4
E. 5

PS03551.01

Attachment:
2019-09-21_1528.png

The number of dots in n*n array of dots = $$n^2$$, and in k*k = $$k^2$$, so $$n^2-k^2=48$$, where n>k.

$$n^2-k^2=48....(n-k)(n+k)=48$$, so product of n-k and n+k has to be 48 and since the product is even, at least one of n-k or n+k should be even.
But n-k and n+k will have the same property, so both have to be even...
check for even numbers whose product is 48..

(a) 48=2*24
so n-k=2 and n+k=24..Add both, so 2n=26..n=13 and k=11
(b) 48=4*12
so n-k=4 and n+k=12..Add both, so 2n=16..n=8 and k=4
(c) 48=6*8
so n-k=6 and n+k=8..Add both, so 2n=14..n=7 and k=1

3 cases

C
_________________
##### Most Helpful Community Reply
SVP
Joined: 20 Jul 2017
Posts: 1504
Location: India
Concentration: Entrepreneurship, Marketing
WE: Education (Education)
Let n and k be positive integers with k ≤ n. From an n × n array of do  [#permalink]

### Show Tags

21 Sep 2019, 03:45
12
2
gmatt1476 wrote:

Let n and k be positive integers with k ≤ n. From an n × n array of dots, a k × k array of dots is selected. The figure above shows two examples where the selected k × k array is enclosed in a square. How many pairs (n, k) are possible so that exactly 48 of the dots in the n × n array are NOT in the selected k × k array?

A. 1
B. 2
C. 3
D. 4
E. 5

PS03551.01

Attachment:
2019-09-21_1528.png

Number of dots within nxn array and NOT in kxk array = n^2 - k^2

So, n^2 - k^2 = 48
—> (n-k)(n+k) = 48

Number of possible pairs of (n,k) = Number if even factors of 48 [Note that both n-k and n+k have to be even, else we will get n or k in fraction values which is not allowed, E.g: factor 1x48 is not allowed, since n-k = 1 and n+k = 48 will give (n,k)= (24.5, 23.5) WHICH IS NOT POSSIBLE]

So, feasible factors of 48 = 2x24 [(n,k) = (13,11)], 4x12 [(n,k) = (8,4)] and 6x8 [(n,k) = (7,1)]
= 3 Possible pairs

IMO Option C

Pls Hit kudos if you like the solution

Posted from my mobile device
##### General Discussion
Manager
Joined: 09 Nov 2018
Posts: 93
Schools: ISB '21 (A)
Re: Let n and k be positive integers with k ≤ n. From an n × n array of do  [#permalink]

### Show Tags

19 Oct 2019, 04:45
gmatt1476 wrote:

Let n and k be positive integers with k ≤ n. From an n × n array of dots, a k × k array of dots is selected. The figure above shows two examples where the selected k × k array is enclosed in a square. How many pairs (n, k) are possible so that exactly 48 of the dots in the n × n array are NOT in the selected k × k array?

A. 1
B. 2
C. 3
D. 4
E. 5

PS03551.01

Attachment:
2019-09-21_1528.png

Hey gmatt1476,
Could you please upload the Offcial solutions for the Quant part also please??
Please... Thanks
VP
Joined: 14 Feb 2017
Posts: 1371
Location: Australia
Concentration: Technology, Strategy
GMAT 1: 560 Q41 V26
GMAT 2: 550 Q43 V23
GMAT 3: 650 Q47 V33
GMAT 4: 650 Q44 V36
GMAT 5: 650 Q48 V31
GMAT 6: 600 Q38 V35
GMAT 7: 710 Q47 V41
GPA: 3
WE: Management Consulting (Consulting)
Re: Let n and k be positive integers with k ≤ n. From an n × n array of do  [#permalink]

### Show Tags

27 Oct 2019, 16:10
1
Alternatively you can trial and error:

n^2 -k^2 = 48
n^2 needs to be greater than 48, so start testing numbers from 7 onward.
7^2 -k^2 =48
-k^2=-1
k^2=1
k=1 .... that's one

Test 8, then 13.
_________________
Here's how I went from 430 to 710, and how you can do it yourself:
https://www.youtube.com/watch?v=KGY5vxqMeYk&t=
Manager
Joined: 25 Sep 2018
Posts: 65
Re: Let n and k be positive integers with k ≤ n. From an n × n array of do  [#permalink]

### Show Tags

06 Nov 2019, 01:05
Dillesh4096 wrote:
gmatt1476 wrote:

Let n and k be positive integers with k ≤ n. From an n × n array of dots, a k × k array of dots is selected. The figure above shows two examples where the selected k × k array is enclosed in a square. How many pairs (n, k) are possible so that exactly 48 of the dots in the n × n array are NOT in the selected k × k array?

A. 1
B. 2
C. 3
D. 4
E. 5

PS03551.01

Attachment:
2019-09-21_1528.png

Number of dots within nxn array and NOT in kxk array = n^2 - k^2

So, n^2 - k^2 = 48
—> (n-k)(n+k) = 48

Number of possible pairs of (n,k) = Number if even factors of 48 [Note that both n-k and n+k have to be even, else we will get n or k in fraction values which is not allowed, E.g: factor 1x48 is not allowed, since n-k = 1 and n+k = 48 will give (n,k)= (24.5, 23.5) WHICH IS NOT POSSIBLE]

So, feasible factors of 48 = 2x24 [(n,k) = (13,11)], 4x12 [(n,k) = (8,4)] and 6x8 [(n,k) = (7,1)]
= 3 Possible pairs

IMO Option C

Pls Hit kudos if you like the solution

Posted from my mobile device

Ok. So why n+k & n-k should be even?
SVP
Joined: 20 Jul 2017
Posts: 1504
Location: India
Concentration: Entrepreneurship, Marketing
WE: Education (Education)
Let n and k be positive integers with k ≤ n. From an n × n array of do  [#permalink]

### Show Tags

Updated on: 07 Dec 2019, 22:36
1
shaonkarim wrote:
Dillesh4096 wrote:
gmatt1476 wrote:

Let n and k be positive integers with k ≤ n. From an n × n array of dots, a k × k array of dots is selected. The figure above shows two examples where the selected k × k array is enclosed in a square. How many pairs (n, k) are possible so that exactly 48 of the dots in the n × n array are NOT in the selected k × k array?

A. 1
B. 2
C. 3
D. 4
E. 5

PS03551.01

Attachment:
2019-09-21_1528.png

Number of dots within nxn array and NOT in kxk array = n^2 - k^2

So, n^2 - k^2 = 48
—> (n-k)(n+k) = 48

Number of possible pairs of (n,k) = Number if even factors of 48 [Note that both n-k and n+k have to be even, else we will get n or k in fraction values which is not allowed, E.g: factor 1x48 is not allowed, since n-k = 1 and n+k = 48 will give (n,k)= (24.5, 23.5) WHICH IS NOT POSSIBLE]

So, feasible factors of 48 = 2x24 [(n,k) = (13,11)], 4x12 [(n,k) = (8,4)] and 6x8 [(n,k) = (7,1)]
= 3 Possible pairs

IMO Option C

Pls Hit kudos if you like the solution

Posted from my mobile device

Ok. So why n+k & n-k should be even?

Let take a case when not both are NOT even
If you see the highlighted part,
A possible value of (n+k)(n-k) = 48x1
—> n + k = 48 &
n - k = 1

Adding both we get (n + k) + (n - k) = 48 + 1
—> 2n = 49
—> n = 24.5
So, we will get the values of n and k as fractions which are not possible as we are talking about nxn matrix. We can’t have a 24.5x24.5 matrix, can we?

Hope I’m clear.

Originally posted by Dillesh4096 on 06 Nov 2019, 01:11.
Last edited by Dillesh4096 on 07 Dec 2019, 22:36, edited 1 time in total.
Manager
Joined: 25 Sep 2018
Posts: 65
Re: Let n and k be positive integers with k ≤ n. From an n × n array of do  [#permalink]

### Show Tags

06 Nov 2019, 01:26
Now its clear! Thanks ☺

Posted from my mobile device
Manager
Joined: 18 Aug 2017
Posts: 130
Re: Let n and k be positive integers with k ≤ n. From an n × n array of do  [#permalink]

### Show Tags

12 Nov 2019, 07:14
Hi Experts,

Is the OA include all possibilities?

I find that (n,k) = (8,4) is also a possible answer.
But it is not in the OA.

If it is further included, then there will we 3+1 = 4 possible pairs.

Please help

Thank you.
Math Expert
Joined: 02 Aug 2009
Posts: 8757
Re: Let n and k be positive integers with k ≤ n. From an n × n array of do  [#permalink]

### Show Tags

12 Nov 2019, 07:44
ballest127 wrote:
Hi Experts,

Is the OA include all possibilities?

I find that (n,k) = (8,4) is also a possible answer.
But it is not in the OA.

If it is further included, then there will we 3+1 = 4 possible pairs.

Please help

Thank you.

It is included in the 3 cases
13,11 ; 8,4 ; 7,1
_________________
CEO
Joined: 03 Jun 2019
Posts: 3230
Location: India
GMAT 1: 690 Q50 V34
WE: Engineering (Transportation)
Re: Let n and k be positive integers with k ≤ n. From an n × n array of do  [#permalink]

### Show Tags

12 Nov 2019, 08:58
1
gmatt1476 wrote:

Let n and k be positive integers with k ≤ n. From an n × n array of dots, a k × k array of dots is selected. The figure above shows two examples where the selected k × k array is enclosed in a square. How many pairs (n, k) are possible so that exactly 48 of the dots in the n × n array are NOT in the selected k × k array?

A. 1
B. 2
C. 3
D. 4
E. 5

PS03551.01

Attachment:
2019-09-21_1528.png

n^2 - k^2 = 48
(n+k)(n-k) = 48 = 2^4*3
n+k = 48; n-k = 1; n = 49/2; k = 47/2; Not feasible
n+k = 24; n-k =2; n = 13; k = 11; Feasible solution
n+k = 12; n-k = 4; n = 8; k = 4; Feasible solution
n+k = 16; n-k = 3; n=19/2; k = 13/2; Not feasible
n+k = 8; n-k = 6; n=7; k = 1; Feasible solution

Since k<n; 3 solutions are feasible

IMO C
_________________
Kinshook Chaturvedi
Email: kinshook.chaturvedi@gmail.com
Intern
Joined: 26 Oct 2019
Posts: 5
Re: Let n and k be positive integers with k ≤ n. From an n × n array of do  [#permalink]

### Show Tags

14 Nov 2019, 10:13
k<=n
when k=1 n=3 ( 8 dots not included)
when k=3 n=5 ( 16 dots not included)
when k=5 n=7 (24 dots not included)

The total number of dots not included = 8+16+24= 48
Answer: 3 pairs of (n,k)

ANS C
Target Test Prep Representative
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 11083
Location: United States (CA)
Re: Let n and k be positive integers with k ≤ n. From an n × n array of do  [#permalink]

### Show Tags

13 Dec 2019, 20:00
3
gmatt1476 wrote:

Let n and k be positive integers with k ≤ n. From an n × n array of dots, a k × k array of dots is selected. The figure above shows two examples where the selected k × k array is enclosed in a square. How many pairs (n, k) are possible so that exactly 48 of the dots in the n × n array are NOT in the selected k × k array?

A. 1
B. 2
C. 3
D. 4
E. 5

PS03551.01

Attachment:
2019-09-21_1528.png

We can create the equation:

n x n - k x k = 48

n^2 - k^2 = 48

(n + k)(n - k) = 48

Since n and k are positive integers (and n ≥ k), n + k and n - k are also positive integers. Since the product of n + k and n - k is 48, both n + k and n - k are factors of 48, with n + k being the larger factor and n - k being the smaller factor. Therefore, we can have the following cases:

1) n + k = 48 and n - k = 1

2) n + k = 24 and n - k = 2

3) n + k = 16 and n - k = 3

4) n + k = 12 and n - k = 4

5) n + k = 8 and n - k = 6.

It seems there are 5 possible ordered pairs of (n, k); however, we have to make sure that n and k are integers. On the other hand, since their sum and difference are integers, we need only to make sure one of the values of n and k is an integer. Let’s solve for n in each set of equations above. By adding the two equations in the cases above, we have:

1) 2n = 49 → n = 49/2

2) 2n = 26 → n = 13

3) 2n = 19 → n = 19/2

4) 2n = 16 → n = 8

5) 2n = 14 → n = 7

We see that only 3 of the 5 cases yield an integer value for n; therefore, there are only 3 ordered pairs of (n, k) that are integer values.

Answer: C
_________________

# Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com

214 REVIEWS

5-STARS RATED ONLINE GMAT QUANT SELF STUDY COURSE

NOW WITH GMAT VERBAL (BETA)

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

Intern
Joined: 25 Nov 2019
Posts: 48
Re: Let n and k be positive integers with k ≤ n. From an n × n array of do  [#permalink]

### Show Tags

13 Dec 2019, 20:22
I did this the long way and not practical for the test. I took differences of squares
1-7 yes
2-nothing =48
...
4-8=48
11-13=48
Admittedly hard to tell when to stop. only stop when k^2-(k-1)^2>48 I guess

Posted from my mobile device
Intern
Joined: 26 Jun 2019
Posts: 24
Re: Let n and k be positive integers with k ≤ n. From an n × n array of do  [#permalink]

### Show Tags

30 May 2020, 10:03
gmatt1476 wrote:

Let n and k be positive integers with k ≤ n. From an n × n array of dots, a k × k array of dots is selected. The figure above shows two examples where the selected k × k array is enclosed in a square. How many pairs (n, k) are possible so that exactly 48 of the dots in the n × n array are NOT in the selected k × k array?

A. 1
B. 2
C. 3
D. 4
E. 5

PS03551.01

Attachment:
2019-09-21_1528.png

48 dots are not to be selected from a n*n and the balance number of dots has to be a square number-> This is how I understood the question.
Basically- n^2-48=k^2. I was able to get just 2 values- 49-48=1^2 and 64-48=4^2.
What is the other possible solution.
Re: Let n and k be positive integers with k ≤ n. From an n × n array of do   [#permalink] 30 May 2020, 10:03

# Let n and k be positive integers with k ≤ n. From an n × n array of do

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

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