I learned the following proposition (in which there is no proof) in a GRE math preparation book. I don't understand what it means and I am not able to find any theorem about this statement in Hardy's An Introduction to the Theory of Numbers.
For any positive integer $c$, the statement $a\equiv b \pmod n$ is equivalent to the congruences $a\equiv b,b+n,b+2n,\dots,b+(c-1)n\pmod {cn}.$
I cannot even apply this proposition to an example such as $7\equiv 1\pmod 6$. If the above is true, then
$7\equiv 1,7,13,19\pmod{24}$ which is obvious not true.
Is there any typo here? Or how should I understand this "proposition"?
Edit: This question may be related to the question here.
Added:
How should I prove this proposition?