$6 \equiv 6 \pmod{7}$ or $6 \equiv -1 \pmod{7}$?

Question

Just a quick question: Is $6 \equiv 6 \pmod{7}$ or $6 \equiv -1 \pmod{7}$?

I'm just trying to figure out whether there was an error in the answer by Jazzachi in this question.

Thanks.

They're both true. This follows from the raw definition of congruence. — Randall, Aug 27 '18 at 01:35
Does $7$ divide $6-6=0$? Yes. Does $7$ divide $6- ^-1 = 7$? Also yes. — Randall, Aug 27 '18 at 01:36
$-1\mod 7$ represents the additive inverse of $1$ modulo $7$ which is $6$ as $1+6=7$ and thus $(1+6)\mod 7=0$. — blub, Aug 27 '18 at 01:37

score 1 · Accepted Answer · answered Aug 27 '18 at 01:45

1

$$6 \equiv 6 \pmod{7}$$ and $$6 \equiv -1 \pmod{7}$$ are both true.

As you know the definition is $$a\equiv b\pmod {n} \iff n|(b-a)$$

For example $$35 \equiv 30 \pmod{5}$$ because $$5| 30-35$$

answered Aug 27 '18 at 01:45

1 Answers1