Definition:Simultaneous Congruences/Linear
< Definition:Simultaneous Congruences(Redirected from Definition:Simultaneous Linear Congruences)
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Definition
A system of simultaneous linear congruences is a set of linear congruences:
- $\forall i \in \closedint 1 r : a_i x \equiv b_i \pmod {n_1}$
That is:
\(\ds a_1 x\) | \(\equiv\) | \(\ds b_1\) | \(\ds \pmod {n_1}\) | |||||||||||
\(\ds a_2 x\) | \(\equiv\) | \(\ds b_2\) | \(\ds \pmod {n_2}\) | |||||||||||
\(\ds \) | \(\cdots\) | \(\ds \) | ||||||||||||
\(\ds a_r x\) | \(\equiv\) | \(\ds b_r\) | \(\ds \pmod {n_r}\) |
Solution
A solution of a system of simultaneous congruences is a residue class modulo $\lcm \set {n_1, n_2, \ldots, n_r}$ such that any element of that class satisfies all the congruences.
Also see
- Results about simultaneous linear congruences can be found here.