## The Definition of Congruence in the Modular Systems

This is the fourth part of the Introduction to the Modular Number Systems Series. In the previous parts, we have learned intuitively the modular systems using a 12-hour analog clock, performed operations with its numbers and introduce the symbol for congruence,  and discussed the different number bases.  In this post, we formally define congruence.

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Recall that the statement $17 \equiv 5 (\mod 12)$ means that 17 gives a remainder of 5 when divided by 12, or that 17 and 5 give the same remainder when divided by 12. We have also learned that 17, 29, and 41 are congruent since all of them give the same remainder (that is 5) when divided by 12. Notice also that since all of them are congruent, » Read more