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\end{align}</math>
{{anchor|Properties}}
The congruence relation satisfies all the conditions of an [[equivalence relation]]:
* Reflexivity: {{math|''a'' ≡ ''a'' (mod ''m'')}}
* Symmetry: {{math|''a'' ≡ ''b'' (mod ''m'')}} if {{math|''b'' ≡ ''a'' (mod ''m'')}}.
* Transitivity: If {{math|''a'' ≡ ''b'' (mod ''m'')}} and {{math|''b'' ≡ ''c'' (mod ''m'')}}, then {{math|''a'' ≡ ''c'' (mod ''m'')
If {{math|''a''<sub>1</sub> ≡ ''b''<sub>1</sub> (mod ''m'')}} and {{math|''a''<sub>2</sub> ≡ ''b''<sub>2</sub> (mod ''m'')}}, or if {{math|''a'' ≡ ''b'' (mod ''m'')}}, then:<ref>{{cite book |author1=Sandor Lehoczky |author2=Richard Rusczky |editor=David Patrick |title=the Art of Problem Solving |year=2006 |isbn=0977304566 |pages=44 |edition=7 |language=en| volume=1|publisher=AoPS Incorporated }}</ref>
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