Congruence Modulo Integer/Examples/8 not equiv 3 mod 2
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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 $\text {4-1}$