Congruence Modulo Integer/Examples/12,345,678,987,654,321 equiv 0 mod 12,345,679
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Example of Congruence Modulo an Integer
- $12 \, 345 \, 678 \, 987 \, 654 \, 321 \equiv 0 \pmod {12 \, 345 \, 679}$
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:
- $12 \, 345 \, 678 \, 987 \, 654 \, 321 - 0 = 12 \, 345 \, 678 \, 987 \, 654 \, 321 = 999 \, 999 \, 999 \times 12 \, 345 \, 679$
Thus:
- $12 \, 345 \, 678 \, 987 \, 654 \, 321 \equiv 0 \pmod {12 \, 345 \, 679}$
$\blacksquare$
Sources
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {4-1}$ Basic Properties of Congruences: Exercise $7 \ \text{(b)}$