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We say that two integers $a$ and $b$ are congruent modulo $m$ if $a − b$ is divisible by $m$. We denote this by $a≡b \pmod m$.

Example 1: $−31 ≡ 11 \pmod 7$

$11 \pmod 7$ is $4$, is it not? $-31 \neq 4$ last time I checked.

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    You're confusing two different (but related) uses of "mod", as a binary *operation* vs. equivalence *relation.* See [this post](http://math.stackexchange.com/a/121054/242) for further explanation.2012-11-28

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Hint: trying subtracting $11$ from $-31$: $\quad -31 - 11 = -42 = -6\cdot 7$.

That is, $7|-42$.

$a\equiv b \pmod m$ by definition means $m|(a - b)$.

Hence, in the case at hand, $-31\equiv 11 \pmod{7}.$

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    Oh. Well then, thank you.2012-11-28