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

Question

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

KSBGC · Accepted Answer

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.

Abhishek009 · Answer

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

broall · Answer

(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

ScottTargetTestPrep · Answer

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

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

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GMAT Problem Solving (PS) Questions

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

Prep Toolkit

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