Sliding Line

If $k=0$ , then $x=y=0$ and $0\in\mathbb{Z}$ , so we have at least one solution. By substituting 0 for $x$ and $y$ , we see that the y-intercept is equal to $k$ , and the x-intercept is equal to $\frac{k}{ \sqrt{2}}$ . For the x-intercept to be an integer, k must be a multiple of $\sqrt{2}$ . However, when k is a multiple of $\sqrt{2}$ , the y-intercept must be irrational, since $\sqrt{2}$ is irrational. Thus, there are no solutions when $k\neq0$ , so the answer is 1.