# Could anyone explain how my textbook gets this modulu congruence statement?

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.

• You're confusing two different (but related) uses of "mod", as a binary operation vs. equivalence relation. See this post for further explanation. Nov 28, 2012 at 3:49

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, $-31\equiv 11 \pmod{7}.$

• Where did 71 come from? I know $7 | -42$, but how does -31 EQUAL 11 mod 7? Nov 28, 2012 at 2:20
• That's not $71$, it's $7|-42$. I removed the space between "7" and "divides" Nov 28, 2012 at 2:24
• -31 is not EQUAL to 11 mod 7; -31 is CONGRUENT to 11, mod 7. Nov 28, 2012 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 $m$. Nov 28, 2012 at 2:36
• Oh. Well then, thank you. Nov 28, 2012 at 2:45