TITLE: Tell whether $(1234657)! +1 \equiv_{11111} (7654321)! -1$ is true or false
QUESTION [6 upvotes]: I have to tell if the following is true or false:
$$(1234657)! +1 \equiv_{11111} (7654321)! -1$$
so by definition we can rewrite the previous equivalence as:
$(1 \cdot 2 \cdot \ldots \cdot 11110 \cdot 11112 \cdot \ldots \cdot 1234567) \cdot 11111 +1 \equiv_{11111} (1 \cdot 2 \cdot \ldots \cdot 11110 \cdot 11112 \cdot \ldots \cdot 7645321) \cdot 11111 -1$
that is 
$a \cdot 11111 +1 \equiv_{11111} b \cdot 11111 - 1$ 
therefore 
$0+1 \equiv_{11111} 0 - 1 $ 
hence however 
$ 1 \not\equiv_{11111} 11110$
So the answer to the question is false. Is my reasoning correct?

REPLY [3 votes]: Yes. It is perfect.
$$ \begin{align} \end{align} $$