Thue's Theorem

If $n>1$, $(a,n)=1$ (i.e., $a$ and $n$ are Relatively Prime), and $m$ is the least integer $>\sqrt{n}$, then there exist an $x$ and $y$ such that

$$ay\equiv \pm x\ \left({{\rm mod\ } {n}}\right)$$

where $0<x<m$ and $0<y<m$.

References

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 161, 1993.