Is this statement about an element in $\mathbb{Z}/n\mathbb{Z}$ correct?

Question

I have the following question.

We have $l\geq 1$ an integer and we consider the cyclic group $(\mathbb{Z}/l\mathbb{Z},\times)$. Given $n\in \mathbb{Z}$ we write $[n]=n+l\mathbb{Z}$ for the class of n. We have also a composition law given as $$[m]\times [n]=[m\times n]:=[mn]$$

Now in a subexercise I need to prove a certain statement. I somewhere used that $[ly]=[0]$ for some $y\in \mathbb{Z}$. But I'm not sure if this is correct. I wanted to prove this claim saying that $ly-0\in l\mathbb{Z}$ but is this enough to say that $$[ly]=[0]$$

Thank you for your help.

Answer 1

Yes. By definition, $$ [0]=0+l\mathbb{Z}=l\mathbb{Z}=ly+l\mathbb{Z}=[ly]\ . $$