Importance of modular arithmetic O-Number Theory Congruences

The mathematician Gauss, who is considered one of the largest in history has said "mathematics is the queen of the sciences and number theory is the queen of mathematics."

Several important discoveries of elementary number theory and Fermat's little theorem, Euler's theorem, Chinese remainder theorem, and many real life situations are based on the simple arithmetic of the remains.

This residue arithmetic is called Modular Arithmetic or Congruences.

My goal initially is to explain "modular arithmetic (Congruences)" in a way so simple that it can be a common man can understand.

L idea is explained with examples from everyday life.

For students who are studying elementary number theory, in its low-grade or graduate courses, this article will serve as a simple introduction.

Modular arithmetic (congruence) of elementary number theory:

We know, we were taught to divide that:

Dividend = Quotient * Divisor + Remainder.

Call the dividend = n, the divisor = q, the ratio = c and the rest = r, then:

n = cq + r or n = r + multiples of q

or if you remove some multiple of q, which gives n = r

What is the point where I want to go, the fact of removing some multilocular of a number to get a new one I have any practical meaning? Let's see.



Example 1:

Analyzed as follows:

Suppose that today is Tuesday. What day is going to be within 300 days from now?

How to solve the above problem?

We get multiples of 7 from 300. We are interested in what's left after removal of the mutiple of 7. We know that 300 / 7 gives the ratio of 42, and the remainder is 6 (since 300 = 42 x 7 + 6) We are not interested in the way that multiples are removed, we are not interested in the ratio but the residue.

Get 6 when some multiples (42) of 7 are separated from 300.

So the question, "What day is within 300 days from now?" now becomes: "What day is 6 days from now?" Because today is Tuesday, 6 days from now will be on Saturday.

The point is, when we are interested in removing the multiples of 7,

300 and 6 means the same to us.

Mathematically, we write this as 300 ≡ 6 (mod 7) and reads like 300 is congruent to 6 modulo 7.

The equation 300 ≡ 6 (mod 7) is called a congruence.

Here called Module 7 and the process is called modular arithmetic.



Example 2:

Suppose you are the 7 am What time will it be 77 hours from now? We have to remove the multiples of 24 from 70. 77 ÷ 24 gives a remainder of 5. or in other words 77 ≡ 5 (mod 24).

Therefore, the time of 77 hours from now is the same as the time of 5 hours from now. 7 am + 5 hours = 24:00 o'clock

. In subsequent articles delve a little more about this beautiful realm of number theory.

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