Congruence Modulo Integer/Examples/12321 equiv 111 mod 3
Jump to navigation
Jump to search
Example of Congruence Modulo an Integer
- $12 \, 321 \equiv 111 \pmod 3$
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:
- $12 \, 321 - 111 = 12 \, 210$
The digit sum of $12 \, 210$ is $3$.
By Divisibility by 3, it follows that:
- $12 \, 210 = k \times 3$
for some $k \in \Z$.
The result follows by definition of congruence.
$\blacksquare$
Sources
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {4-1}$ Basic Properties of Congruences: Exercise $7 \ \text{(e)}$