Finding $-8$ in $\Bbb Z \pmod 5$

Question

This is a very simple question but what is $-8$ in $\Bbb Z \pmod 5$? I am looking for a positive number. I know $5 \times 2 = 10$ so it is either $2$ or $-3$. I assume I am just looking at it a little confused because of the negative value.

Answer 1

$$-8 = 5\cdot (-2) + 2$$

The answer is 2

Answer 2

Just keep adding or subtracting $5$ until you get the appropriate value:

$$-8 + 5 = -3 \\ -3 + 5 = 2.$$

Answer 3

There are various ways, e.g.

$(1)\ \ \ {-}8 \equiv -8+5+5\equiv 2$

$(2)\ \ \ {-}8 \equiv -4 -4 \equiv 1 + 1$

$(3)\ \ \ {-}8 \equiv 8(-1)\equiv 3(-1)\equiv 2$