Statement #1 tells us that \((x^3)(y^3) = 441^3\). This can be mathematically simplified down to \((x)(y)=441\). Therefore, both \(x\) and \(y\) are factors of \(441\).

Factoring \(441\) isn't hard, especially once we realize that \(441\) is not only divisible by \(9\), but \(441 = 450-9\).

Factoring a common \(9\) out of this expression gives us:

\(450-9 = 9(50-1) = 9*49 = 3*3*7*7\)

\(x\) or \(y\) could be any combination of these prime values. \(y\) could be equal to 3 (not a multiple of 7), and it could also be equal to 7 (obviously, a multiple of 7.) Statement #1 is clearly insufficient.

Statement #2 is profoundly insufficient by itself. It tells us nothing about \(y\), so it is easy to eliminate.

As we combine the statements together, we can now ask ourselves the question, "does it matter that \(x\) is a single digit integer?" If \(x\) must be a single digit integer, there are only four possible solutions: \(x\)\(=1,3,7,\) or \(9\).

Given these four values for \(x\), here are the possible values for \(x\) and \(y\) (notice that there is no reason to actually multiply them out, since the problem only wants us to find out if \(y\) is a multiple of 7.)

(

\(x\)) *(

\(y\))

\(1\) * (

\(3*3*7*7\))

\(3\)*(

\(3*7*7\))

\(7\)*(

\(3*3*7\))

\(9\)*(

\(7*7\))

In every case, \(y\) must be a multiple of 7. Combining these two statements together is sufficient.

The answer is C.
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Aaron J. Pond

Veritas Prep Elite-Level Instructor

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