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Applications of Number Theory

Number theory is widely applicable now a days not in the field of mathematics but also in the field of scientific researches.
Number Theory includes the different aspects of natural numbers and their extensions in various fields of mathematics and science. Number theory can be subdivided into Algebraic Number Theory, Combinatorial Number Theory, Analytic Number Theory and Computational Number Theory etc.

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Applications in Real Life: 
Number theory can be used to find out some of the important divisibility tests, whether a given integer n is divisible by an integer m, which are required in various fields of banking, credit card securities, e – commerce websites etc. To find out interesting puzzles as various everyday product identification codes, etc., to get the knowledge of various bar codes, their combinations, making of modular designs etc. 
Let n be a fixed positive integer. For any integers a and b we define a congruent to b modulo n, if n divides a – b. 
The congruence modulo n is an equivalence relation on the set of integers.

Various applications of congruence to find divisibility are:
An integer n is divisible by 10 if and only if an integer m is divisible by 10, written as n congruent to m modulo (10).
Because n is congruent to m modulo (10), so n is divisible by 5 if and only if m is divisible by 5.
Because 10 is congruent to 0 modulo (2), 10i is congruent to 0i modulo (2), for all positive integers iso n is divisible by 2i if and only if m is divisible by 2i.

Chinese Remainder Theorem:
This theorem is widely applied in abstract algebra and other area of researches also. It states that, if (m1, m2) = 1, then, 
x1 congruent to a1 mod m1, and xcongruent to a2 mod m2,….(1)
Have a common solution which is unique modulo m1* m2. Further if m1r + m2s = 1, where r and s are integers, then x is congruent to (a2m1r + a1m2s) modulo (m1m2is the unique solution of (1).


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Solved Examples:
Example 1: Solve the congruence 7x congruent to 3 modulo 5.
Solution: We will find x such that $\frac{5}{7x-3}$… (i)
We see that x = 4 is the first positive integer such that (i) is true. 
Similarly, 4 + 5 = 9, 9 + 5 = 14, etc. all satisfy (i). 
x = 4 - 5 = -1, -1 - 5 = -6, -6 - 5 = -11 etc also satisfy (i). 
Hence, the solution set of the given congruence is  S = {…., -11, -6, -1, 4, 9, 14, 19,….}

Example 2: Find the remainder when 250 is divided by 7.
Solution: We know, 23 is congruent to 1 modulo 7, this implies
(2316 is congruent to 116 modulo 7, this implies
248 is congruent to 1 modulo 7…..(i)
Also, we know, 22 is congruent to 22 modulo 7(ii)
From (i) and (ii), we get 250 is congruent to 4 modulo seven and hence, 4 is the required remainder.

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