Solution
= Solution
The <linear Diophantine equation solvability criterion> says that $ax+by=c$ has an integer solution exactly when $\gcd(a,b)$ divides $c$. The <Euclidean algorithm> gives
$$
\begin{aligned}
2314&=2(1007)+300,\\
1007&=3(300)+107,\\
300&=2(107)+86,\\
107&=86+21,\\
86&=4(21)+2,\\
21&=10(2)+1.
\end{aligned}
$$
Thus $\gcd(1007,2314)=1$, which divides $37$. Therefore
$$
\boxed{1007x+2314y=37\text{ has integer solutions}.}
$$