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So I am working on a relation problem on the set of positive integers if and only if $a \equiv b \pmod n$. That in itself is fine.

But when I want to prove that the relation is for example reflexive, I am trying to do so using the definition of congruence mod $n$. So for $a-a = kn$, is there an integer $k$ that makes this hold? So that would be $0$, but then I started doubting if that holds, since $n$ must be greater than $1$ and positive. Must the factor $n$ also be greater than $1$ and positive (or on the set of positive integers)? Or can it be any integer?

5xum
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spectr1
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1 Answers1

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$0$ is an integer. The condition for congruence only speaks of integers, not positive integers or integers greater than $1$ or anything like that.

The definition is as follows:

$$a\equiv b\pmod n\iff \exists k\in\mathbb Z: a-b=k\cdot n$$

This condition is satisfied if $a=b$ and $k=0$.

5xum
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