Yes, I have seen that this question has been asked and answered before in this same website, but answers given there were mostly in regards to geometry, or non-mathematical examples (e.g. "the 'e's in the word 'between' are congruent, but not equal", "two triangles with the same dimensions and points are equal").
I can understand this just fine, but I can't use this advice when it comes to pure numbers, like mods. For example: $17 \equiv 5 (mod(6))$. How is this statement correct? If we solve $5 (mod(6))$ we get $5$, so that would mean $17 \equiv 5$.
Also, $17 \equiv 4(mod(13))$, which means $17 \equiv 4$, and $17 \equiv 3(mod(7))$, which is $17 \equiv 3$
So then, $17 \equiv 5 \equiv 4 \equiv 3$ is true? That doesn't seem right. Are all positive integers congruent to one another?
I know this may seem like a simple matter to some, but I'm seriously stuck.