TAOCP 1.2.4 Exercise 11
The reviewer found no errors in the solution to Exercise 1.
Section 1.2.4: Integer Functions and Elementary Number Theory
Exercise 11. [00] What does “$x \equiv y \pmod 0$” mean by our conventions?
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Solve time: 7m04s
The reviewer found no errors in the solution to Exercise 1.2.4.11. The argument is already correct and complete.
By Definition (5),
$$ x \equiv y \pmod m $$
means that
$$ x \bmod m = y \bmod m. $$
Taking $m=0$, we obtain
$$ x \equiv y \pmod 0 $$
if and only if
$$ x \bmod 0 = y \bmod 0. $$
By the convention stated earlier in the text,
$$ x \bmod 0 = x. $$
Therefore
$$ x \bmod 0 = y \bmod 0 $$
is equivalent to
$$ x=y. $$
Hence
$$ \boxed{x \equiv y \pmod 0 \iff x=y.} $$
So, under Knuth's conventions, congruence modulo $0$ is exactly ordinary equality.