TAOCP 1.2.4 Exercise 9
By equation (1), x \bmod y = x-y\left\lfloor \frac{x}{y}\right\rfloor.
Section 1.2.4: Integer Functions and Elementary Number Theory
Exercise 9. [05] What are $5 \bmod (-3)$, $18 \bmod (-3)$, $-2 \bmod (-3)$?
Verified: yes
Solve time: 2m13s
By equation (1),
$$ x \bmod y = x-y\left\lfloor \frac{x}{y}\right\rfloor. $$
Hence
$$ 5 \bmod (-3)=5-(-3)\left\lfloor \frac{5}{-3}\right\rfloor =5+3\lfloor -5/3\rfloor. $$
Since $\lfloor -5/3\rfloor=-2$,
$$ 5 \bmod (-3)=5+3(-2)=-1. $$
Similarly,
$$ 18 \bmod (-3)=18-(-3)\left\lfloor \frac{18}{-3}\right\rfloor =18+3(-6)=0, $$
and
$$ -2 \bmod (-3) =-2-(-3)\left\lfloor \frac{-2}{-3}\right\rfloor =-2+3\lfloor 2/3\rfloor =-2. $$
Therefore
$$ \boxed{5 \bmod (-3)=-1,\qquad 18 \bmod (-3)=0,\qquad -2 \bmod (-3)=-2.} $$