  We will study the solution of a congruence, $x \equiv g^{(1/2)\omega_g(2^n)}
\bmod 2^n$, depending on the integers $g$ and $n$, where $\omega_g(2^n)$
denotes the order of $g$ modulo $2^n$. Moreover, we introduce an application of
the above result to the study of an estimation of exponential sums.
