If a and b are integers and |a - b| = 16, what is the minimum possible

Question

If a and b are integers and |a - b| = 16, what is the minimum possible value of ab?

A. -16
B. -32
C. -48
D. -64
E. -80

Chethan92 · Accepted Answer

Given |a-b| = 16
Possible values are (-15,1) (-14,2)......(-8,8)
Here the min value is obtained when the value of a and b are same, but with opposite signs.
a = -8 and b = 8
ab = -64
D is the answer.

devavrat · Answer

Hi just want to understand why cant it be -20,4??
once the value is in mod can't it be written as |20-4|=16??
This would give the answer as -80
Am I missing something over hear??

nick1816 · Answer

if you consider a=20 and b=-4 then |a-b|=24 
or if you consider a=-20 and b=4, even then |a-b|=24
Hence ab can never be equal to -80

ScottTargetTestPrep · Answer

If a = 8 and b = -8 (or, a = -8 and b = 8), we see that |a - b| = 16 and the product ab = -64. Had we choose any other pairs of numbers for a and b, the product would be greater than -64. For example, if a = 7 and b = -9 (or, a = -9 and b = 7), ab = -63. If a = 6 and b = -10 (or a = -10 and b = 6), ab = -60. We see that both -63 and -60 are greater than -64. So -64 is the smallest product for ab.

Answer: D

SarmadIK · Answer

But when a=-4 and b=-20, dont we get |a-b|=16 since the negative sign of the equation would cancel the negative sign of b and make it -4+20?

nick1816 · Answer

If a=-4 and b=-20, value of |a-b| will be equal to 16. But product of ab will be equal to 80, not -80.

Deepender · Answer

Why not C ?can any one explain as -48 I less than -64

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Amit0701 · Answer

Because in a number line the more you move towards left side the smaller the number is.
-64 is on left side of -48. hence -64 is smaller than -48.
So D will be the answer
(how we got D is already explained by others in above comments)

Shrey08 · Answer

The minimum possible value of  ab  will be the maximum possible value with a  -ve  sign.

ab  will be negative if either  a or b  is negative.

Since either  a or b  is negative  |a - b| = 16  represents the sum of  a & b = 16 
Now for a given sum of 2 numbers the product is maximum when the numbers are same.

Therefore,  a=b; a+a=16; 2a=16; a=8 
The product is  64 
Again, the minimum possible value of  ab   will be the maximum possible value with a  -ve  sign.

-64 , option D

Fdambro294 · Answer

The Absolute Value of: |a - b| = 16 can be interpreted as the following:

The Distance on the Number Line between Unknown Variable a and Unknown Variable b is equal to exactly 16 units.

Thus, for example, if a (or b) is equal to 2, then the other Variable must be 16 units away at either -14 or +18.

Concept: given the sum of 2 unknown numbers, we can maximize the Magnitude/Size of the Product by making those numbers equal.

In this case, we can minimize the value by making the negative product’s size as large as possible. This can be done by placing the 2 values equally apart from Zero —— a = -8 and b = +8

Answer: minimum product of ab = -8 * +8 = -64

This is the largest magnitude of (-)negative number that we can get given the absolute value expression

D

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ashishpathak · Answer

I am not able to get it, if we take numbers -20 and -4 the -20-(-4)=-16 am I missing something and mod -16 is 16 therefore should not E be the answer.

Shrey08 · Answer

We are asked to find out minimum value of the product of  a  and  b . With the numbers you've picked the value of product of  a  and  b  would be (-20)*(-4) = 80. Clearly, this is not the minimum value we can get.

check here  https://gmatclub.com/forum/if-a-and-b-are-integers-and-a-b-16-what-is-the-minimum-possible-293469.html#p2591291  for detailed explanation.

Shubhradeep · Answer

Clearly we need one of a and b as a negative integer and the other one a positive integer.

So we shall only select the regions where a and b have different signs i.e Quadrants 2 and 4.
Now both the lines have a slope of 1, i.e, a and b will increase or decrease with difference of 1.
For example, {a=-8, b=8}; {a=-9, b=7}; {a=-10, b=6} like that.
For the sake of understanding, we shall take a and b both to be positive and ab>0. Later on we 
can make one of them negative in order to make ab<0 and calculate the minimum possible value
of ab.

So now we are considering a and b both to be positive and we are looking for the maximum possible
value of ab.  
We can sense that when the difference between the absolute values of a and b are smaller, value of 
ab gets higher. 
For example, 8*8=64  
 9*7=63  
 10*6=60

so, we know our potential absolute values of a and b will be 8 and 8. It can be {a= -8,b=8} or {a=8,b= -8}
So ab=64.

This is a very common trick for these kind of questions. We have to make the 2 variables as close as 
possible to make their product value maximum. In this case, we just had to tweak the sign of a variable.

So, Correct answer option is D

akshatv45 · Answer

We know that AM >= GM

(a + (-b))/2 >= √(a)(-b)

=> ab >= -((a-b)/2)^2
=> ab >= -(16/2)^2
=> ab >= -64

Answer : D

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If a and b are integers and |a - b| = 16, what is the minimum possible

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If a and b are integers and |a - b| = 16, what is the minimum possible

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