Congruence Modulo Integer/Examples/12,345,678,987,654,321 not equiv 0 mod 12,345,678

Example of Non-Congruence Modulo an Integer

$12 \, 345 \, 678 \, 987 \, 654 \, 321 \not \equiv 0 \pmod {12 \, 345 \, 678}$

Proof

We have that $12 \, 345 \, 678$ ends in $8$.

Thus by Divisibility by 2, $12 \, 345 \, 678$ has $2$ as a divisor.

But $12 \, 345 \, 678 \, 987 \, 654 \, 321$ ends in $1$.

Thus, also by Divisibility by 2, $12 \, 345 \, 678 \, 987 \, 654 \, 321$ does not have $2$ as a divisor.

By Divisor Relation is Transitive, it follows that $12 \, 345 \, 678$ is not a divisor of $12 \, 345 \, 678 \, 987 \, 654 \, 321$.

The result follows by definition of congruence and divisibility.

$\blacksquare$

Sources

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