Why does $-14 \bmod 12 = 10$? I would be grateful if someone could explain this to me step by step, for I am but a novice in the field of modular arithmetic. [edit] I obtained this equation by playing around with values for (x,y,z) in: x mod y = z on the Google calculator. Thank you!
Edit: I would like to hit myself on the head- I see it now. Thanks to everyone who responded! Unless there is a discussion going on in this post right now, it can be marked as resolved.
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Because $10 -(-14) = 24$ which is a multiple of 12. In general, $ a\equiv b \pmod m $ means that $m$ divides $a-b$.
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I know it's tempting to think of mod as an operation, but most often in modular arithmetic, it's a modifier of $\equiv$, specifying exactly what "congruence" means. Really, if the notation had been $-14\equiv_{12}10$ instead, this would've been so much clearer.
So with this interpretation in mind, $-14\mod 12$ doesn't mean anything. On the other hand, $-14\equiv10\mod12$ is a statement. We can check whether it's true by using the definition: $$ a\equiv b\mod c\iff c\mid a-b $$ In this case, we get $12\mid -14-10$, which is true. This means that $-14\equiv 10\mod 12$ is true.
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If you have $a\text{ mod }b = c$ then there exists $k\in\mathbb Z$ such that $a=kb+c$. In your case, you have $a=-14$, $b=12$ and $c=10$ while $k=-2$: $$ -14 = -2\cdot 12+10. $$
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-24 is the largest number less than -14 which is congruent to 0 mod 12. -24 + 10 = -14 so -14 = 10 mod 12.
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Whenever you get a positive number, you remove $12$ until you get a number between $0$ And $11$ to get the modulo.
Example: $\ 37\bmod{12}\equiv 37-12\equiv 25\equiv 25-12\equiv 13\equiv 13-12\equiv 1\pmod{12}$
For a negative number, in the same way just add $12$ until you get something positive.
Example: $\ {-}14\bmod{12}\equiv -14+12\equiv -2\equiv -2+12\equiv 10\pmod{12}$
$-14\equiv10\pmod{12}\,$ because $\,{-}14=10-12\cdot 2$. In other words, $-14$ and $10$ leaves the same remainder after dividing $12.$