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Which one of the following is the minimum value of the sum of two

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Which one of the following is the minimum value of the sum of two  [#permalink]

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New post 27 Nov 2016, 11:32
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Question Stats:

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Which one of the following is the minimum value of the sum of two integers whose product is 36?
(A) 37
(B) 20
(C) 15
(D) 13
(E) 12

Source: Nova GMAT
Difficulty Level: 550

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Re: Which one of the following is the minimum value of the sum of two  [#permalink]

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New post 27 Nov 2016, 13:43
SajjadAhmad wrote:
Which one of the following is the minimum value of the sum of two integers whose product is 36?
(A) 37
(B) 20
(C) 15
(D) 13
(E) 12


a*b = 36

If a = b = 6 ; then ab = 36

a + b = 12 ( which is among the given option )

Hence, answer will be (E) 12

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Re: Which one of the following is the minimum value of the sum of two  [#permalink]

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New post 27 Nov 2016, 18:36
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SajjadAhmad wrote:
Which one of the following is the minimum value of the sum of two integers whose product is 36?
(A) 37
(B) 20
(C) 15
(D) 13
(E) 12


\((a-b)^2 \geq 0 \iff (a+b)^2 \geq 4ab \iff a+b \geq 2\sqrt{ab} \quad \forall a,b>0\)

Hence \(a+b \geq 2\sqrt{ab} =2\sqrt{36} =12 \iff a=b=6\)

The answer is E

Mathematically, AM-GM inequality is defined that: \(x+y \geq 2\sqrt{xy} \quad \forall x,y \geq 0\)
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Re: Which one of the following is the minimum value of the sum of two  [#permalink]

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New post 01 Oct 2019, 00:06
SajjadAhmad wrote:
Which one of the following is the minimum value of the sum of two integers whose product is 36?
(A) 37
(B) 20
(C) 15
(D) 13
(E) 12

Source: Nova GMAT
Difficulty Level: 550


SajjadAhmad,

I think it's a flawed question. why ?

let's integers be x and y.

Their sign is not stated in the question. Thus they both can be negative or both can be positive.

xy = 36

1 x 36 = 36

2 x 18 = 36

3 x 12 = 36

4 x 9 = 36

6 x 6 = 36.

Since we don't know their sign , both x and y could be negative . Ultimate product is positive.

-1 x -36 = 36

Minimum Value : -1 + (-36) = -37.

This option is not included in the question .

Thnks.
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Re: Which one of the following is the minimum value of the sum of two  [#permalink]

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New post 07 Oct 2019, 20:04
1
SajjadAhmad wrote:
Which one of the following is the minimum value of the sum of two integers whose product is 36?
(A) 37
(B) 20
(C) 15
(D) 13
(E) 12

Source: Nova GMAT
Difficulty Level: 550



We have the following positive integers that yield a product of 36.

1 and 36

2 and 18

3 and 12

4 and 9

6 and 6

Thus, the minimum sum is 6 + 6 = 12.

Alternate Solution:

It is a well-known fact that for a given area of a rectangle, the smallest perimeter is obtained when the shape is a square. We can use this fact, noting that xy = 36 is an area formula, where x = length and y = width. Thus, since a square has length and width equal, we see that x = y, so we can have x^2 = 36, and so x = 6. Thus, the minimum sum (half-perimeter) will be 6 + 6 = 12.

Answer: E
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Re: Which one of the following is the minimum value of the sum of two   [#permalink] 07 Oct 2019, 20:04
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