Answer
If ${\bf{F}}$ is a gradient vector field, then the line integral of ${\bf{F}}$ along every closed curve is zero.
Work Step by Step
If ${\bf{F}}$ is a gradient vector field such that ${\bf{F}} = \nabla f$, where $f$ is some potential field for ${\bf{F}}$, then by definition ${\bf{F}}$ is conservative. By Theorem 1, the line integral of ${\bf{F}}$ around a closed curve is zero. So, we must add the word "closed" to the statement so that it becomes true.
Thus, the true statement should be:
If ${\bf{F}}$ is a gradient vector field, then the line integral of ${\bf{F}}$ along every closed curve is zero.