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