If x and y are integers and |x - y| = 12, what is the minimum possible

Hi All,

Sometimes the answer choices to a given question provide a big 'hint' as to how you can go about solving it. This prompt can also be solved without any complex math ideas - you just need to do a bit of 'brute force' math and you'll have the answer relatively quickly.

We're told that X and Y are INTEGERS and |X - Y| = 12. We're asked for the MINIMUM possible value of (X)(Y).

Since all of the answer choices are NEGATIVE, this tells us that ONE of the two variables MUST be negative (and the other must be positive), so we should restrict our work to those options.

IF...
X = 11, Y = -1, then XY = -11
X = 10, Y = -2, then XY = -20
X = 9, Y = -3, then XY = -27
X = 8, Y = -4, then XY = -32
X = 7, Y = -5, then XY = -35
X = 6, Y = -6, then XY = -36
X = 5, Y = -7, then XY = -35

From this, we can conclude the XY will start to get bigger as X continues to decrease down to 1, so there's no need to do any additional work.

Final Answer:

GMAT assassins aren't born, they're made,
Rich

I'm not sure what is going on!
Why would you say maximum value when the question says minimum value of xy is to be found.
And fyi the minimum value is -11 which is not even there in the answer choices

Hey Bunuel
Is the question wrong? It says find the minimum and the two answers above find the maximum. Besides the minimum value (-11) isnt even there in the answer choice.
Am I missing something here?

Regards
Parth

The minimum value is -36, which is less tha -11.

\({-11}>{-36}\) on the number line. Thus, D is the correct answer.

Oh yea I'm sorry for this.
Was doing the question at 2am in the morning so maybe all the numbers scrambled up!

But thanks for the clarification.

Regards

hello Bunuel

not sure but |24-12|=12 and then xy =24(-12) =-288

I might not be getting the question can you help

Thanks
Utkarsh

Hi Utkarsh,

In your example, X = +24 and Y = +12, so the product is (24)(12) = +288. From the answer choices, we know that the minimal product is NEGATIVE, so +24 and +12 cannot be the combination of numbers that you're looking for. If you read through my explanation (a few posts up the page), then you'll see how to quickly get to the correct answer.

GMAT assassins aren't born, they're made,
Rich

We see that all the answer choices are negative; thus, one of the values of x and y must be negative and the other positive. We can let x be negative, and y be positive.

If x = -1, then y = 11 and xy = -11.
If x = -2, then y = 10 and xy = -20.
If x = -3, then y = 9 and xy = -27.
If x = -4, then y = 8 and xy = -32.
If x = -5, then y = 7 and xy = -35.
If x = -6, then y = 6 and xy = -36.
If x = -7, then y = 5 and xy = -35.

We can stop here since we see that we have the minimum product of -36 when x = -6 and y = 6. (Note: Had we continued, the product will be increasing rather than decreasing since the product will be a mirror image of what we have. That is, after the last product -35, the subsequent ones will be -32, -27, -20 and -11.)

Answer: D

Approached the question by observing that x and y must be numbers such that their difference must be 12. Found the possible factor sets of the numbers given in the options. For example factor sets of 12 would be (1x-12), (-1x12), (2x-6), (-2x6), (-3x4) and (3x-4). The difference of none of the numbers in each sets results in 12.
Followed the same approach for other options as well.
Only Option D i.e. -36 has one possible factor set (6,-6) whose difference results in 12. Hence it is the right option.

Is there some place I can learn about this concept?

"...when the sum of two numbers is given, the max product is when both x and y are same numeric value...
so xy will have max numeric value when numeric value of x and y=12/2=6..."

Very confused as to why we're looking at the max product when they're asking for the minimum and how that provides the answer?

Hi salphonso,

This is a rare issue on the GMAT (and you will probably not see it on Test Day), but the idea appears more often in Geometry questions, in situations in which you are are trying to maximize the area of a shape with two dimensions (such as a square or rectangle).

For example, if the length + the width of a rectangle = 10, then the maximum possible area would occur when L = W = 5.

In this prompt, we're dealing with one negative and one positive number - and we're looking for the minimum possible product, so we want the one that is 'most negative.' In terms of an Absolute Value, we're looking for the one that's 'largest', which is why this math concept applies. We hit that exact value with the values +6 and -6.

GMAT assassins aren't born, they're made,
Rich

That clears it up, thanks Rich!

if we pick a number, wouldn't we have to make sure that it fulfills both absolute value equations?
such as x-y = 12 and x-y = -12 (if this is even possible, probably not)

I've came up with 6/-6 as well but I thought those numbers are not usable because it wouldnt fulfill the x-y=-12 equation.

Any advice?

Hi chrtpmdr,

When you're dealing with an equation that includes an Absolute Value, you have to consider that there will likely be MORE than one solution. You're NOT trying to find a solution that fits two equations - you're trying to find all the solutions that fit ONE Absolute Value equation.

With |X - Y| = 12, there are LOTS of potential solutions.

Two of those solutions are:
X = +6 and Y = -6
X = -6 and Y = +6

With either of these options, you will have the answer to the question (re: What is the minimum value of (X)(Y)?)

GMAT assassins aren't born, they're made,
Rich

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