modulo arithmetic

In the Third Century B.C.E., Euclid formalized, in his book Elements, the fundamentals of arithmetic, as well as showing his lemma, which he used to prove the Fundamental theorem of arithmetic. Euclid's Elements also contained a study of Perfect numbers in the 36th proposition of Book IX. Diophantus of Alexandria wrote Arithmetica, containing 130 equations and treating the essence of problems having only one solution, fraction or integer.
Congruence relation
Modular arithmetic can be handled mathematically by introducing a congruence relation on the integers that is compatible with the operations of the ring of integers: addition, subtraction, and multiplication. For a positive integer n, two integers a and b are said to be congruent modulo n, written: if their difference a − b is an integer multiple of n (or n divides a − b). The number n is called the modulus of the congruence.
For example, because 38 − 14 = 24, which is a multiple of 12.
The same rule holds for negative values: Equivalently, can also be thought of as asserting that the remainders of the division of both and by are the same. For instance: because both 38 and 14 have the same remainder 2 when divided by 12. It is also the case that is an integer multiple of 12, which agrees with the prior definition of the congruence relation.
A remark on the notation: Because it is common to consider several congruence relations for different moduli at the same time, the modulus is incorporated in the notation. In spite of the ternary notation, the congruence relation for a given modulus is binary. This would have been clearer if the notation a ≡n b had been used, instead of the common traditional notation.
The properties that make this relation a congruence relation (respecting addition, subtraction, and multiplication) are the following.
If

and then:
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It should be noted that the above two properties would still hold if the theory were expanded to include all real