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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$\begingroup$ You're confusing two different (but related) uses of "mod", as a binary operation vs. equivalence relation. See this post for further explanation. $\endgroup$– Bill DubuqueNov 28, 2012 at 3:49
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1 Answer
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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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$\begingroup$ Where did 71 come from? I know $7 | -42 $, but how does -31 EQUAL 11 mod 7? $\endgroup$ Nov 28, 2012 at 2:20
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$\begingroup$ That's not $71$, it's $7|-42$. I removed the space between "7" and "divides" $\endgroup$– amWhyNov 28, 2012 at 2:24
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$\begingroup$ -31 is not EQUAL to 11 mod 7; -31 is CONGRUENT to 11, mod 7. $\endgroup$– amWhyNov 28, 2012 at 2:32
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$\begingroup$ To quote your post: "We say that two integers a and b are congruent modulo m if a − b is divisible by m"... $a\equiv b \pmod m$ reads $a$ is congruent to $b$, modulo $m$. $\endgroup$– amWhyNov 28, 2012 at 2:36
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