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If a, b are integers and |a-b|=8, which of the following is the smalle

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If a, b are integers and |a-b|=8, which of the following is the smalle [#permalink] New post   14 Jul 2016, 21:33
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If a, b are integers and |a-b|=8, which of the following is the smallest possible value of ab?
A. -16
B. 0
C. 16
D. 7
E. 40

*An answer will be posted in 2 days.
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ShreyasCM
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Re: If a, b are integers and |a-b|=8, which of the following is the smalle [#permalink] New post   14 Jul 2016, 21:49
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The value will be lowest when a= -4 and b= 4--> | -4 - 4| = 8
(an)min = 4*-4 = -16
Hence A.

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Re: If a, b are integers and |a-b|=8, which of the following is the smalle [#permalink] New post   16 Jun 2017, 22:16
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yk510 wrote:
Why is the answer not 0? If a=8 and b=0?


Hi

The question asks, "What is the smallest possible value of a*b?"

Like you, when I read the question, I also noticed that getting '0' is easy: take a=8 and b=0. But then I saw that there is another option given which is less than 0 (which is -16). So we have to check whether -16 is also possible or not, since the question asks for lowest value possible.

So with some trial/error, I saw that indeed -16 is possible when I take a=-4 and b=4. because |-4-4| = |-8| = 8, and -4*4 = 16.
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Re: If a, b are integers and |a-b|=8, which of the following is the smalle [#permalink] New post   16 Jun 2017, 23:21
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In min-max expression questions,
we have to take the maximum value of one integer
and a minimum value of another integer which satisfy the equation
Since the expression is |a-b| = 8
the range of this expression must be -8 <= a - b <= 8

This is possible when a=-4 and b=4 or a=4,b=-4
In either case, ab = -16(Option A)
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Re: If a, b are integers and |a-b|=8, which of the following is the smalle [#permalink] New post   22 Aug 2020, 23:58
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Notice that |a-b|=8 is literally saying two numbers who's distance is 8 from each other.
This can happen anywhere in the number line.

If both a and b are positive then ab>0
If both a and b are negative then ab>0
But if one is negative and the other positive the product ab<0

So we are shooting for a and b of opposite signs.
a=-7, b=1, ab=-7
a=-6, b=2, ab=-12
a=-5, b=3, ab=-15
a=-4, b=4, ab=-16
a=-3, b=5, ab=-15
.
.
we can see when we choose 4 and -4 the product is minimized to -16
Answer: A
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Re: If a, b are integers and |a-b|=8, which of the following is the smalle [#permalink] New post   18 Jul 2016, 09:11
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a=4, b=-4. Hence, the answer is A.
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Re: If a, b are integers and |a-b|=8, which of the following is the smalle [#permalink] New post   16 Jun 2017, 14:36
Why is the answer not 0? If a=8 and b=0?
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Re: If a, b are integers and |a-b|=8, which of the following is the smalle [#permalink] New post   20 Jun 2017, 12:11
yk510 wrote:
Why is the answer not 0? If a=8 and b=0?


When you convert this expression to liner equation it is a+(-b) = 8 when a>b or a + (-b) = -8 when a<b.

So, now if you see the values of a and b for which the product wil be minimum when corresponding values are 4 and -4 and hence the product = -16. Hope this makes sense
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Re: If a, b are integers and |a-b|=8, which of the following is the smalle [#permalink] New post   21 Jun 2020, 13:50
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Min / Max value of a sum of a integer is when both the digits are same. So max product or min product will occur when x.y = Same digit,
|x-y|= 8, so x, y should be 4 ,
x= 4, y=-4 or x= -4, y= 4 :As we are looking for min value so one of x,y has to be negative to get min value.

4*-4 = -16
Answer : A
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Re: If a, b are integers and |a-b|=8, which of the following is the smalle [#permalink] New post   13 Jul 2020, 08:36
yk510 wrote:
Why is the answer not 0? If a=8 and b=0?


Also, check for the negative values as one of the options is a negative integer.
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If a, b are integers and |a-b|=8, which of the following is the smalle [#permalink] New post   Updated on: 23 Aug 2020, 00:28
MathRevolution wrote:
If a, b are integers and |a-b|=8, which of the following is the smallest possible value of ab?
A. -16
B. 0
C. 16
D. 7
E. 40

*An answer will be posted in 2 days.


The smallest possible value of \(ab \) will be the largest possible value with a \(−ve\) sign.

\(ab\) will be negative if either \(a\) or \(b\) is negative.

Since either \(a\) or \(b\) is negative \(|a−b|=8\) represents the sum of \(a & b\)
Now for a given sum of 2 numbers the product is maximum when the numbers are same.

Therefore, \(a=b;a+a=8;2a=8;a=4\)
The product is \(16\)
Again, the smallest possible value of \(ab\) will be the largest possible value with a \(−ve\) sign.

\(−16\), option A

Originally posted by Shrey08 on 17 Aug 2020, 10:18.
Last edited by Shrey08 on 23 Aug 2020, 00:28, edited 2 times in total.
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Re: If a, b are integers and |a-b|=8, which of the following is the smalle [#permalink] New post   05 Nov 2022, 04:04
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