For how many integers pair (x,y) satisfies the result

Question

For how many integers pair (x,y) satisfies the result 
(1/x)+((1/y)=1/12

a) 12
b) 6
c) 10
d) 16
e) 32

Kindly excuse me for not able to type the question in proper format as its my first time I posted a question , hope next question will be in proper format.
thanks

Celestial09 · Answer

Is anyone up for the correct answer along with simple explanation too?
thanks

kinsho · Answer

It seems that the accepted answer is wrong, according to WolframAlpha. The answer should be 29, which isn't even listed as a selectable answer.

Furthermore, this also looks like an equation that represents a hyperbola, which is something I've never seen tested before on the GMATs. I am truly curious where the OP pulled this problem.

pranavmimani · Answer

What is the solution ? Detailed explanation, please. According to me it can be satisfied by a pair such as (3, -4) only. (1/x ) will be 1/3 and (1/y) shall be (-1/4). Solving the fractions will give us 1/12. I have used the factor pairs of 12 to satisfy the equation.

davedekoos · Answer

First of all, I would be very surprised if this was an official GMAT problem. Even though the equation represents a hyperbola, which isn't tested on the GMAT in terms of geometry, I think the question is still valid in terms of algebra, even if the scope is still beyond what the GMAT would likely test. To solve this, we could test every number from -infinity to infinity, but that might take a while.

Instead, we can rework the expression into something that is more useful. \frac{1}{x}+\frac{1}{y}=\frac{1}{12} \frac{1}{y} = \frac{1}{12}-\frac{1}{x} = \frac{x-12}{12x} y=\frac{12x}{x-12} So for y to be an integer, x-12 will have to be a factor of 12x. And we have to remember also that we can't use (x,y) = (0,0) because of the original form of the equation. We can cut down on our work also, be realizing that there are two symmetries in this equation. The plot of the equation is symmetric about y=x, meaning that for every (x,y) = (a,b), the reverse, (b,a) will also be a solution. There are also asymptotes at x=12 and y=12, meaning that the function is symmetric about the point (12,12). So knowing those symmetries, if we find all the values of x between 0 and 12 that give integer solutions to the function, we can multiply that by 4 to find the total number of solutions. (If this all sounds very confusing, it's because it is. There is a plot of the function below to help clarify). To find the values of x that give an integer solution, we don't even need to calculate all the values of y, we can use our rearranged equation y=\frac{12x}{x-12} , and look for every x such that x-12 is a factor of 12 (1, 2, 3, 4, 6, 12), or a factor of x, or a combination of both. It is still a little time consuming, but with this approach we can find all the values of x between 0 and 12 that satisfy our conditions: x y 3 -4 4 -6 6 -12 8 -24 9 -36 10 -60 11 -132 There are 7 (x, y) integer pairs that satisfy the equation for x between 0 and 12. Terrific, we multiply by 4 and we have 28. But we're still missing one! Notice (0, 0) has been left out. Even though it satisfies the equation y=\frac{12x}{x-12} , it would violate the original form of the equation (we're not allowed to divide by 0!). But it has a symmetric point at (24, 24) that IS allowed. So the total number of solutions is 29. Just as kinsho pointed out. There are some other shortcuts we could have taken, but this is already so beyond the scope of the GMAT that it's becoming silly to continue to work on it. For those interested, the plot of the function looks like this: 1 on x and 1 on y is 1 on 12.png

vitaliyGMAT · Answer

As davedekoos said this is the equation of a hyperbola but there is one distinct feature that put’s it into another category, that is x,y should be integers. It’s still a hyperbola but now we are interested in finding out integer solutions and that equation becomes a Diophantine equation.

Now let’s look at this question from the perspective of number theory.

1/x + 1/y = 1/12
12x + 12y = xy
12x + 12y – xy =0

We need to factorize it and for that we subtract 144 from left and right parts.

12x – xy +12y – 144 = -144 
(x-12)(12-y) = -144
(x-12)(y-12) = 144

Now we need to represent 144 as a product of 2 factors and find out our pairs of x and y.

(1*144) (2*72) (3*48) (4*36) (6*24) (8*18) (9*19) and 12*12.

All factor pairs (except 12*12) can be written in reverse order, so we have total 15 positive solutions.

Because we have product, x and y can also take negative values. Using same logic we got 15 negative solutions, but in this case we need to discard (-12)*(-12), because it will give us x=0, y=0. In this point the original equation is undetermined (we can’t divide by zero).

Hence we have total number of integer pairs 15 + 14 = 29

I don’t know why this option is absent. This question looks more like AIME or AMC type of questions.

For how many integers pair (x,y) satisfies the result

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