TAOCP 1.2.4 Exercise 8
By definition (1), $x \bmod y = x - y \lfloor x/y \rfloor$ when $y \ne 0$, and $x \bmod 0 = x$.
Section 1.2.4: Integer Functions and Elementary Number Theory
Exercise 8. [00] What are $100 \bmod 3$, $100 \bmod 7$, $-100 \bmod 7$, $-100 \bmod 0$?
Verified: yes
Solve time: 35s
By definition (1), $x \bmod y = x - y \lfloor x/y \rfloor$ when $y \ne 0$, and $x \bmod 0 = x$. We compute each case separately.
For $100 \bmod 3$, we have $\lfloor 100/3 \rfloor = \lfloor 33.333\ldots \rfloor = 33$, so
$100 \bmod 3 = 100 - 3 \cdot 33 = 100 - 99 = 1.$
For $100 \bmod 7$, we have $\lfloor 100/7 \rfloor = \lfloor 14.2857\ldots \rfloor = 14$, so
$100 \bmod 7 = 100 - 7 \cdot 14 = 100 - 98 = 2.$
For $-100 \bmod 7$, we have $\lfloor -100/7 \rfloor = \lfloor -14.2857\ldots \rfloor = -15$, so
$-100 \bmod 7 = -100 - 7 \cdot (-15) = -100 + 105 = 5.$
Finally, $-100 \bmod 0 = -100$ by definition.
Thus, the results are
$\boxed{100 \bmod 3 = 1,\quad 100 \bmod 7 = 2,\quad -100 \bmod 7 = 5,\quad -100 \bmod 0 = -100}.$