Answer
If $a|b$ and $b|a$, where $a$ and $b$ are integers, then $a=b$ or $a=-b$.
Work Step by Step
Let $a|b$ and $b|a$, where $a, b\in \mathbb{Z}$. Because $a$ factors $b$, there exists some integer $c$ s.t. $a\cdot c=b$. Because $b|a$, there exists an integer $d$ s.t. $b\cdot d=a$. Combining these equations, we see that $a\cdot c=b\cdot d \cdot c=b$. This shows that the product of $b$ and $d$ is equal to 1; because they are integers, this further shows that $|b|=|d|=1$ Either both numbers are positive, or both are negative. If they are positive, then $a\cdot 1=b$, so $a=b$. If they are negative, then $a\cdot -1=b$, so $a=-b$. Thus concludes the proof for the statement above.
