Congruent Numbers are not necessarily Equal

Theorem

Let $x, y, z \in \R$ be real numbers such that:

$x \equiv y \pmod z$

where $x \equiv y \pmod z$ denotes congruence modulo $z$.

Then it is not necessarily the case that $x = y$.

Proof

Proof by Counterexample:

We have that:

$11 - 5 = 6 = 3 \times 2$

and so by definition of congruence modulo $2$:

$10 \equiv 4 \pmod 2$

But $11 \ne 5$.

$\blacksquare$

Also see

• Equal Numbers are Congruent

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {4-1}$ Basic Properties of Congruences: Example $\text {4-3}$