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
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 6$. The Residue Classes: Example $8$