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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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.

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), 10^{i} is congruent to 0^{i} modulo (2), for all positive integers i, so n is divisible by 2^{i} if and only if m is divisible by 2^{i}.

This theorem is widely applied in abstract algebra and other area of researches also. It states that, if (m_{1}, m_{2}) = 1, then,

x_{1} congruent to a_{1} mod m_{1}, and x_{2 }congruent to a_{2} mod m_{2},….(1)

Have a common solution which is unique modulo m_{1}* m_{2}. Further if m_{1}r + m_{2}s = 1, where r and s are integers, then x is congruent to (a_{2}m_{1}r + a_{1}m_{2}s) modulo (m_{1}m_{2}) is the unique solution of (1).

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,….}

Hence, the solution set of the given congruence is S = {…., -11, -6, -1, 4, 9, 14, 19,….}

(2^{3}) ^{16} is congruent to 1^{16} modulo 7, this implies

2^{48} is congruent to 1 modulo 7…..(i)

Also, we know, 2^{2} is congruent to 2^{2} modulo 7…(ii)

From (i) and (ii), we get 2^{50} is congruent to 4 modulo seven and hence, 4 is the required remainder.