Definition:Simultaneous Congruences/Linear

From ProofWiki
Jump to navigation Jump to search


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:

\(\ds a_1 x\) \(\equiv\) \(\ds b_1\) \(\ds \pmod {n_1}\)
\(\ds a_2 x\) \(\equiv\) \(\ds b_2\) \(\ds \pmod {n_2}\)
\(\ds \) \(\cdots\) \(\ds \)
\(\ds a_r x\) \(\equiv\) \(\ds b_r\) \(\ds \pmod {n_r}\)


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.