Modular Arithmetic

Congruence of Numbers and Set of Residues

The concept of modular arithmetic was developed by Carl Friedrich Gauss. Modular arithmetic is extensively used in number theory. Let us learn more about this.

Congruence of Numbers
Here is a 12-hour analogue clock.

 
In the 12-hour format, the same time is used to represent two different parts of the day (i.e. A.M. and P.M.).
For example:
The second hour of the day (or 02:00 hours) and the fourteenth hour of the day (or 14:00 hours) are both represented by the time 2 o’clock.
Similarly, the seventh hour of the day (or 07:00 hours) and the nineteenth hour of the day (or 19:00 hours) are both represented by the time 7 o’clock.
This is because a time is repeated after every 12 hours on the clock.
This is an example of modulo 12. 
Consider the statement “the fourteenth hour of the day is the same as 2 o’clock”.
In this case, we can say that 14 is congruent to 2, when the repetition occurs after 12 steps.
This relation can be expressed using modulo as 14 congruent to 2 (mod 12).
The given relation can also be expressed symbolically using the symbol “≡”.
The symbol “≡” is used to represent congruent relations. 
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