## Quick Reference

The following are examples of congruence equations:*x*+5 ≡ 3 (mod 7); this has the solution *x* ≡ 5 (mod 7).*x* ≡ 5 (mod 4); this has no solutions.*x*^{2} ≡ 1 (mod 8); this has solutions *x* ≡ 1,3,5 or 7 (mod 8).*x*^{2}+2*x*+3≡ 0 (mod 6); this has solutions *x* ≡ 1 or 3 (mod 6).In seeking solutions to a congruence equation, it is necessary only to consider a complete set of residues and find solutions in this set. The examples (i) and (ii) above are linear congruence equations. The linear congruence equation *ax* ≡ *b* (mod *n*) has a solution if and only if (*a*, *n*) divides *b*, where (*a*, *n*) is the greatest common divisor of *a* and *n*.

*x*+5 ≡ 3 (mod 7); this has the solution *x* ≡ 5 (mod 7).

*x* ≡ 5 (mod 4); this has no solutions.

*x*^{2} ≡ 1 (mod 8); this has solutions *x* ≡ 1,3,5 or 7 (mod 8).

*x*^{2}+2*x*+3≡ 0 (mod 6); this has solutions *x* ≡ 1 or 3 (mod 6).

*Subjects:*
Mathematics.

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