Congruence Modulo Integer/Examples/8 not equiv 3 mod 2

Example of Non-Congruence Modulo an Integer

$8 \not \equiv 3 \pmod 2$

Proof

By definition of congruence:

$x \equiv y \pmod n$ if and only if $x - y = k n$

for some $k \in \Z$.

We have:

$8 - 3 = 5 = 2 \times 2 + 1 = 2 \cdotp 5 \times 2$

and so $8 - 3$ is not of the form $k \times 2$ for some $k \in \Z$.

Thus:

$8 \not \equiv 3 \pmod 2$

$\blacksquare$

Sources

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