Modulo Multiplication is Well-Defined/Examples/2x3 equiv -6x15 mod 4
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Example of Use of Modulo Multiplication is Well-Defined
| \(\ds 2\) | \(\equiv\) | \(\ds -6\) | \(\ds \pmod 4\) | Congruence Modulo $4$: $2 \equiv -6 \pmod 4$ | ||||||||||
| \(\ds 3\) | \(\equiv\) | \(\ds 15\) | \(\ds \pmod 4\) | Congruence Modulo $4$: $3 \equiv 15 \pmod 4$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 2 \times 3 = 6\) | \(\equiv\) | \(\ds \paren {-6} \times 15 = -90\) | \(\ds \pmod 4\) |
To confirm:
| \(\ds 6 - \paren {-90}\) | \(=\) | \(\ds \paren {-24} \times 4\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 6\) | \(\equiv\) | \(\ds -90\) | \(\ds \pmod 4\) | ||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 2\) | \(\ds \pmod 4\) |
That is:
- $\eqclass 2 4 \eqclass 3 4 = \eqclass 2 4$
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 6$. The Residue Classes: Example $8$