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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.

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    $\begingroup$ The equal sign is not used because the numbers are not strictly equal, the numbers fall under the same equivalence relation. When, you talk about the number 1 modulo 5, you are talking about the set of numbers $$\{\dots,-9,-4,1,6,11,17,\dots\}$$ all at once. $\endgroup$ – Jonathan Davidson Jun 24 '17 at 18:45
  • $\begingroup$ So when you use the $\equiv$ you just mean that the two numbers (or expressions) are part of the same set? So, could I also technically write $$1 \equiv 2 \gt 0$$ as a true statement? $\endgroup$ – Joao Eanes Jun 24 '17 at 18:51
  • $\begingroup$ @JoaoEanes How are you interpreting that statement? $\endgroup$ – Jack M Jun 24 '17 at 19:16
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I think that you are misunderstanding the meaning of $\equiv$ in number theory. The statement $$17\equiv5\equiv4\equiv3$$ has no meaning whatsoever. Neither does the statement $17\equiv 4$. The essential part is the $\bmod$ at the end. The statement $$a\equiv b(\bmod c)$$ means "the remainder of $a \div c$ is equal to the remainder of $b \div c$". So when you exclude the "mod", you are refraining from specifying what is being divided by, which is an essential part of the statement.

However, you could say that since $$17 \equiv 4 (\bmod 13)$$ and $$17 \equiv -9 (\bmod 13)$$ then $$4 \equiv -9 (\bmod 13)$$ The modulus is an essential part of the statement, though.

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