Is xy an integer?

Answer C
Only combining both we'll get a integer value of XY

Statement 1:
Nothing about y

Statement 2:
Nothing about x

Combined, when the ratios are multiplied together, the pi from both statements cancel and we are left with an integer.
sufficient

Answer: C

MANHATTAN GMAT OFFICIAL SOLUTION:

The question asks whether a particular product (xy) is an integer. Note that this is a Yes-No question.

Statement 1: NOT SUFFICIENT. This statement only refers to x. Without knowing anything about y, you cannot know whether xy is an integer.

Statement 2: NOT SUFFICIENT. Likewise, this statement only refers to y, so it cannot be sufficient.

Statements 1 and 2 TOGETHER: SUFFICIENT. The super-fancy way to get the answer is to realize that x is a fixed number, completely determined by its definition in the statements. The same is true of y. Why is this the case? All circles are the same shape, so they are all similar to each other. Likewise, all squares are the same shape and are similar to each other. So when you inscribe a circle inside a square (to touch all four sides of the square), there

There should be only one “shape” to the picture in your mind of a square with a circle inscribed inside it, touching all four walls. All that’s different is how large or small that picture is, so the ratio of the square’s area to the circle’s area is fixed:

The same is true for y, the ratio of the circle’s area to the area of an inscribed square:

So x and y are fixed. You don’t know what their values are, but you don’t care: in theory, you could calculate those values. And then you could determine whether the product is an integer or not.

The longer way to get the answer is to actually figure out these ratios.

Take x first. Call the side of the square 1. Then the radius of the inscribed circle is 1/2, and the area of the circle is \(\pi*r^2= \frac{\pi}{4}\). The area of the square is 1^2 = 1, so the ratio of the square’s area to the circle’s area is \(1:\frac{\pi}{4}\), or \(\frac{4}{\pi}\). That’s the value of x.

Now take y. Call the side of the square 1 again. Then the diameter of the circle is the diagonal of the square, which is \(\sqrt{2}\). The radius of the circle is \(\frac{\sqrt{2}}{2}\), and the area of the circle is \(\pi*r^2 = \pi(\frac{\sqrt{2}}{2})^2 = \frac{\pi}{2}\). That’s the value of y, since the area of the square is just 1, and you want the ratio of the circle’s area to the square’s area.

Finally, the product of x and y is \((\frac{4}{\pi})(\frac{\pi}{2}) = 2\), which is indeed an integer.

The correct answer is C.

Main points to know to solve this question:
If a circle is inscribed in a square: the diameter of the circle is equal to side of the square
If a square is inscribed in a circle the diagonal of the square is equal to diameter of the circle( diagonal of square = root2*s)
Assume any value for the side of the square, using both statements the product of xy will be an integer as the pie gets cancelled out