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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 (mod m).

Example 1: −31 ≡ 11 (mod 7)

11 mod 7 is 4, is it not? -31 ≠ 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 for further explanation. –  Bill Dubuque Nov 28 '12 at 3:49
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up vote 6 down vote accepted

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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Where did 71 come from? I know $7 | -42 $, but how does -31 EQUAL 11 mod 7? –  Doug Smith Nov 28 '12 at 2:20
    
That's not $71$, it's $7|-42$. I removed the space between "7" and "divides" –  amWhy Nov 28 '12 at 2:24
    
-31 is not EQUAL to 11 mod 7; -31 is CONGRUENT to 11, mod 7. –  amWhy Nov 28 '12 at 2:32
    
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 $m$. –  amWhy Nov 28 '12 at 2:36
    
Oh. Well then, thank you. –  Doug Smith Nov 28 '12 at 2:45
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