11. Do there exist such integers $x$ and $y$ for which the following equalities hold simultaneously:

$$
x y=4747 \text { and } x-y=-54 ?
$$
11. Solution. The product $x y$ of integers $x$ and $y$ can only equal 4747 in four cases: $1 \cdot 4747, 47 \cdot 101$, $(-1) \cdot(-4747)$, and $(-47) \cdot(-101)$. From the second condition, it follows that $x$ is less than $y$. By checking, we find that both conditions are only satisfied in the cases where $x=47$ and $y=101$ or $x=$ $=-101$ and $y=-47$.