All fun technical language aside, the idea is really simple. If two numbers belong to the same remainder class, then in some way, they are the same. That is, they are the same up to division by \(b\). In the case where \(b = 5\) above, the numbers \(8\) and \(23\text{,}\) while not the same number, are the same when it comes to dividing by 5, because both have remainder \(3\text{.}\)
in-context