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

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Senior Manager
Joined: 04 Sep 2017
Posts: 291
Let n and k be positive integers with k ≤ n. From an n × n array of do  [#permalink]

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21 Sep 2019, 04:30
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Difficulty:

95% (hard)

Question Stats:

40% (02:57) correct 60% (02:50) wrong based on 150 sessions

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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 2053 times ]
Math Expert
Joined: 02 Aug 2009
Posts: 8151
Let n and k be positive integers with k ≤ n. From an n × n array of do  [#permalink]

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19 Oct 2019, 06:17
2
1
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

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

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21 Sep 2019, 04:45
6
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

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: 89
Re: Let n and k be positive integers with k ≤ n. From an n × n array of do  [#permalink]

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19 Oct 2019, 05:45
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,
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Re: Let n and k be positive integers with k ≤ n. From an n × n array of do  [#permalink]

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27 Oct 2019, 17:10
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.
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Manager
Joined: 25 Sep 2018
Posts: 64
Re: Let n and k be positive integers with k ≤ n. From an n × n array of do  [#permalink]

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06 Nov 2019, 02:05
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?
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Let n and k be positive integers with k ≤ n. From an n × n array of do  [#permalink]

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06 Nov 2019, 02:11
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 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.
Manager
Joined: 25 Sep 2018
Posts: 64
Re: Let n and k be positive integers with k ≤ n. From an n × n array of do  [#permalink]

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06 Nov 2019, 02:26
Now its clear! Thanks ☺

Posted from my mobile device
Manager
Joined: 18 Aug 2017
Posts: 102
Re: Let n and k be positive integers with k ≤ n. From an n × n array of do  [#permalink]

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12 Nov 2019, 08:14
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.

Thank you.
Math Expert
Joined: 02 Aug 2009
Posts: 8151
Re: Let n and k be positive integers with k ≤ n. From an n × n array of do  [#permalink]

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12 Nov 2019, 08:44
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.

Thank you.

It is included in the 3 cases
13,11 ; 8,4 ; 7,1
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Re: Let n and k be positive integers with k ≤ n. From an n × n array of do  [#permalink]

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12 Nov 2019, 09:58
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
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Re: Let n and k be positive integers with k ≤ n. From an n × n array of do   [#permalink] 12 Nov 2019, 09:58
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