Modulo Addition is Well-Defined/Examples/2+3 equiv -6+15 mod 4

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Example of Use of Modulo Addition is Well-Defined

We have:

\(\ds 2\) \(\equiv\) \(\ds -6\) \(\ds \pmod 4\) Congruence Modulo $4$: $2 \equiv -6 \pmod 4$
\(\ds 3\) \(\equiv\) \(\ds 15\) \(\ds \pmod 4\) Congruence Modulo $4$: $3 \equiv 15 \pmod 4$
\(\ds \leadsto \ \ \) \(\ds 2 + 3 = 5\) \(\equiv\) \(\ds -6 + 15 = 9\) \(\ds \pmod 4\)


To confirm:

\(\ds 5 - 9\) \(=\) \(\ds 1 \times 4\)
\(\ds \leadsto \ \ \) \(\ds 5\) \(\equiv\) \(\ds 9\) \(\ds \pmod 4\)
\(\ds \) \(\equiv\) \(\ds 1\) \(\ds \pmod 4\)


That is:

$\eqclass 2 4 + \eqclass 3 4 = \eqclass 1 4$


Sources