In the mathematics, the number theory is biggest and also oldest one. The number theory is pure mathematics. The number theory concerns about numbers, that is, whole numbers or rational numbers or fractions. Prime number and prime factorization are very important in the number theory.

Some of the Number theory types are given below.

1. Algebraic Number Theory

2. Greatest common division and least common division

3. Arithmetic Functions

4. Combinatorial Number Theory

5. Binomial Coefficients

6. Quadratic Residues.

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The numbers of functions are divisor function, Riemann zeta function and Quotient function. The number theory may be found on ore in the year of 1988 and Belie (1966).

Algebraic Number Theory is one of the branches of number theory. This deals with algebraic numbers. The algebraic number theory developed for solving problems for basic number theory, i.e. rational numbers and integers. The sum or product of algebraic integer is again an algebraic integer. In general, algebraic numbers are complex form, but they may also be real term.

Arithmetic number theory is a branch of mathematics. It deals with integers or, more generally, number computation. The arithmetic operations are addition, subtraction, factorization, multiplication, and power computation and root operation. Number theory is also called as higher arithmetic function

**MULTIPLICATIVE NUMBER THEORY:**

It is one of the number theories, the multiplication two functions. f (m, n)= f (m) f(n) ; f is a function

**SEQUENCES:**

1, 2, 3, 4, 5....

1, 3, 5, 7, 9....

These patterns are normally known as sequences. An arrangement of numbers of which one number is elected as the first, another as second, and another as third and so on is known as a sequence.

**SERIES: **

** **When the terms of a sequence are connected by the sign +, it is called a series. Thus a_{1} + a_{2} + a_{3} + … + a_{n} + … is an infinite series. The symbol ? an is used to denote a series.