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In the equation n^2 kn + 16 = 0, n is an integer. Which of the

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Bunuel
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In the equation n^2 kn + 16 = 0, n is an integer. Which of the [#permalink] New post   23 Mar 2023, 01:30
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In the equation \(n^2 – kn + 16 = 0\), n is an integer. Which of the following could be the value of k?

I. 8
II. 15
III. –17

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III



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D
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D
$!vakumar.m
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In the equation n^2 kn + 16 = 0, n is an integer. Which of the [#permalink] New post   Updated on: 06 Nov 2023, 06:08
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n^2–kn+16=0; given n is an integer
I. k = 8
n^2 - 8n + 16 =0
(n-4)^2 = 0
=> n = 4 which is an integer. I is Correct
II. k = 15
n^2 - 15n + 16 = 0
sqrt(15^2 -4*1*16) = sqrt161 is not an integer, hence, n is not an integer. II is incorrect.
III. k = –17
n^2 - (-17)n + 16 = 0
n^2 + 17n + 16 = 0
(n+1) * (n+16) = 0
n = -1, n = -16. So n is an integer. III is correct.
Answer D

Originally posted by $!vakumar.m on 23 Mar 2023, 02:42.
Last edited by $!vakumar.m on 06 Nov 2023, 06:08, edited 1 time in total.
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Re: In the equation n^2 kn + 16 = 0, n is an integer. Which of the [#permalink] New post   04 Nov 2023, 14:52
sivakumarm786 wrote:
n^2–kn+16=0; given n is an integer
I. k = 8
n^2 - 8n + 16 =0
(n-4)^2 = 0
=> n = 4 which is an integer. I is Correct
II. k = 15
n^2 - 15n + 16 = 0
sqrt(15^2 -4*1*16) = sqrt161 is not an integer, hence, n is not an integer. II is incorrect.
III. k = –17
n^2 - (-17)n + 16 = 0
n^2 + 17n + 16 = 0
(n+1) * (n+16) = 0
]n = -1, n = 16. So n is an integer. III is correct.
Answer D


I believe it is n = -1 and n = -16 but the answer remains the same !
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Re: In the equation n^2 kn + 16 = 0, n is an integer. Which of the [#permalink] New post   07 Nov 2023, 07:49
We can look at it through the product and sum of roots concept, since K represents sum of the roots

Product of roots= +16
Factors of 16= 16,1 and 2,8 and 4,4

now since the product is positive, both the factors should be of the same sign(either both negative or both positive)

Now checking the values-
8= Possible (4,4)
15= Not possible (because to get 15, 16 and 1 will have to be of opposite sign, making the product negative)
-17= Possible (-16,-1)


Bunuel Kindly confirm my reasoning
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Re: In the equation n^2 kn + 16 = 0, n is an integer. Which of the [#permalink]
07 Nov 2023, 07:49
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