Congruence Modulo Integer/Examples/57 not equiv 208 mod 4

Example of Non-Congruence Modulo an Integer

$57 \not \equiv 208 \pmod 4$

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:

$208 - 57 = 151 = 37 \times 4 + 3$

and so $208 - 57$ is not of the form $k \times 4$ for some $k \in \Z$.

Thus:

$57 \not \equiv 208 \pmod 4$

$\blacksquare$

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {4-1}$ Basic Properties of Congruences: Exercise $7 \ \text{(c)}$