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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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21 Sep 2019, 04:30
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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:
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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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19 Oct 2019, 06:17
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: 20190921_1528.png The number of dots in n*n array of dots = \(n^2\), and in k*k = \(k^2\), so \(n^2k^2=48\), where n>k. \(n^2k^2=48....(nk)(n+k)=48\), so product of nk and n+k has to be 48 and since the product is even, at least one of nk or n+k should be even. But nk 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*24so nk=2 and n+k=24..Add both, so 2n=26..n=13 and k=11 (b) 48=4*12so nk=4 and n+k=12..Add both, so 2n=16..n=8 and k=4 (c) 48=6*8so nk=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
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21 Sep 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: 20190921_1528.png Number of dots within nxn array and NOT in kxk array = n^2  k^2 So, n^2  k^2 = 48 —> (nk)(n+k) = 48 Number of possible pairs of (n,k) = Number if even factors of 48 [Note that both nk 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 nk = 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 solutionPosted from my mobile device




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Re: Let n and k be positive integers with k ≤ n. From an n × n array of do
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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: 20190921_1528.png Hey gmatt1476, Could you please upload the Offcial solutions for the Quant part also please?? Please... Thanks



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Re: Let n and k be positive integers with k ≤ n. From an n × n array of do
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27 Oct 2019, 17:10
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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Re: Let n and k be positive integers with k ≤ n. From an n × n array of do
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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: 20190921_1528.png Number of dots within nxn array and NOT in kxk array = n^2  k^2 So, n^2  k^2 = 48 —> (nk)(n+k) = 48 Number of possible pairs of (n,k) = Number if even factors of 48 [Note that both nk 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 nk = 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 solutionPosted from my mobile deviceOk. So why n+k & nk should be even?



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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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06 Nov 2019, 02:11
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: 20190921_1528.png Number of dots within nxn array and NOT in kxk array = n^2  k^2 So, n^2  k^2 = 48 —> (nk)(n+k) = 48 Number of possible pairs of (n,k) = Number if even factors of 48 [ Note that both nk 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 nk = 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 solutionPosted from my mobile deviceOk. So why n+k & nk should be even? Let take a case when not both are even If you see the highlighted part, A possible value of (n+k)(nk) = 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.



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Re: Let n and k be positive integers with k ≤ n. From an n × n array of do
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06 Nov 2019, 02:26
Now its clear! Thanks ☺
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Re: Let n and k be positive integers with k ≤ n. From an n × n array of do
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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.
Please help
Thank you.



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Re: Let n and k be positive integers with k ≤ n. From an n × n array of do
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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.
Please help
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
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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: 20190921_1528.png n^2  k^2 = 48 (n+k)(nk) = 48 = 2^4*3 n+k = 48; nk = 1; n = 49/2; k = 47/2; Not feasible n+k = 24; nk =2; n = 13; k = 11; Feasible solution n+k = 12; nk = 4; n = 8; k = 4; Feasible solution n+k = 16; nk = 3; n=19/2; k = 13/2; Not feasible n+k = 8; nk = 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
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