If $b\ne 0$ is an integer in $\mathbb{Z}$, which of the following statements is mathematically correct?

Step 1 (Understand the divisibility relation): An integer $b$ divides an integer $a$ (written $b\mid a$) if there exists an integer $k$ such that:

Step 2 (Test divisibility of zero): Let's test the number $a=0$. For any non-zero integer $b\ne 0$, we want to find if there is an integer $k$ such that $0=b\cdot k$.
Choosing $k=0$ (which is a valid integer):

This equation is mathematically true for any non-zero integer $b$. Therefore, $b$ divides $0$, denoted as:

Step 3 (Analyze other options):

• $0\mid b$: Divisibility by zero is undefined.

• $b\mid 1$: Only true if $b=\pm 1$.

• $b\mid (b+1)$: Only true if $b=\pm 1$.

Step 4 (Conclusion): The correct statement is $b\mid 0$, which is at option index 0.