Why is $20 ≡ 2 \pmod 6\;?$

Question

Could anyone explain to me why $20 ≡ -22 \pmod 6\;?$

At school we did the following method to find $-x \mod n$ by doing:

• $x \mod n$ (in this case $22 \mod 6 = 4)$

• $n - r$ (in this case $6-4 = 2$)

• and so the answer to $-22 \mod 6$ is $2$.

So how can it be that $20\equiv 2\pmod 6$?

Answer 1

$$20 = 3\cdot 6 + 2{}{}{}{}{}{}{}$$

Alternatively, let's check this out: $$20 \equiv 2 \pmod 6 \iff 6\mid (20-2) \iff 6\mid 18\;\;\large \color{green}\checkmark$$

Answer 2

$a\equiv b \mod 6$ simply means that $6$ divides the difference between $a$ and $b$.

So $20-(-22)=42=7 \times 6$ and $20-2=18=3\times 6$