What is the smallest positive integer that satisfies the congruence $4x \equiv 13 \pmod{27}$?
Observing that $4 \cdot 7 = 28 = 27 + 1,$ we multiply both sides of the given congruence by 7 to find $28x \equiv 91 \pmod{27}$. Since $28x\equiv x\pmod{27}$ and $91\equiv10 \pmod{27}$, we conclude that $x\equiv 10\pmod{27}$. Therefore, $\boxed{10}$ is the smallest positive integer satisfying the given congruence.