If x and y are positive integers, is 2x a multiple of y?

\(x, y > 0\), integers

"Is \(\frac{2x}{y}\) an integer?"
"Is \(2x\) a multiple of (divisible by) \(y\)?"
"Is \(y\) a factor of \(2x\)?"

Statement 1) This statements says that \(\frac{2x + 2}{y} = \frac{2(x+1)}{y}\) is an integer. In other words, \(2(x+1)\) is a multiple of \(y\).

In order for both \(2x\) and \(2(x+1)\) to be divisible by \(y\), \(y\) must be either \(1\) (factor of \(2\) and both \(x\) and \((x+1)\)) or \(2\) (factor of \(2\)). Thus if \(y\) equals \(1\) or \(2\), then \(\frac{2x}{y}\) is an integer. If \(y\) has any other value under this constraint, \(\frac{2x}{y}\) is not an integer.

You can also break this statement into two expressions: \(\frac{2x + 2}{y} = \frac{2x}{y} + \frac{2}{y}\). If \(\frac{2x}{y} + \frac{2}{y}=integer\), then in order for \(\frac{2x}{y}\) to be an integer, \(\frac{2}{y}\) must also be an integer. Thus if \(y\) equals \(1\) or \(2\), then \(\frac{2x}{y}\) is an integer. If \(y\) has any other value, \(\frac{2x}{y}\) is not an integer.

Insufficient.

Statement 2) This statement says that \(\frac{y}{x}\) is an integer. So \(y = x*k\), where \(k\) is some integer.

In order for \(\frac{2x}{y} = \frac{2*x}{k*x}\) to be an integer, \(\frac{2}{k}\) must be an integer, ie \(k\) must equal either \(1\) or \(2\). Thus if \(y = 2x\) or \(y = x\), \(2x\) will be a multiple of \(y\). With any other value of \(k\), \(2x\) will not be a multiple of \(y\).

Insufficient.

Combined)
From statement 1, \(\frac{2x}{y}\) is an integer when \(y={1,2}\), and not an integer when \(y\) has another value.
From statement 2: \(\frac{2x}{y}\) is an integer when \(y=2x\) or \(y=x\), and not an integer when \(y\) is a different multiple of \(x\) than \(2\).

So under the constraints of both statements, \(\frac{2x}{y}\) is an integer when \((x,y)\) = \((1,1)\), \((2,2)\), or \((1,2)\).

It only takes one example that follows the two constraints but is not one of these 3 possibilities to prove the combined statements insufficient. A couple examples are \((x,y) = (1,4)\), or \((2,6)\).