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Let $a\in\mathbb{Z},n\in\mathbb{N}$. If $a$ has a remainder $r$ when divided by $n$, then $a\equiv r\pmod n$

I've done some of these questions before with modulus and division, but I'm unsure of how to approach it with the addition of a remainder

vonbrand
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user158123
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  • How do you define remainders? How do you define $a \equiv r \pmod n$? Personally, I would define both by saying that there exists some $q \in \mathbb Z$ such that: $$ a = qn + r $$ – Adriano Jun 20 '14 at 02:44

1 Answers1

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The division algorithm tells us that, when an integer $x$ is divided by a natural number $n$, then $x = qn + r$ for some $q \in \mathbb{Z}$ and $0 \leq r < n$.

Now can you show $n|(x-r)$?

By definition, $x \equiv r \pmod{n} \iff n|(x-r)$.

Kaj Hansen
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