These are the most elementary methods in the subject of number theory, and explain the foundations of basic arithmetic operations like addition, multiplication and division between integers. More precisely, these methods offer a study of basic arithmetic laws, their constitution, and explain problems like why addition or multiplication of two positive numbers results in a greater third number, or how division or subtraction between two positive numbers results in a smaller number, or what generalizations can be made when these processes are carried out over numbers of distinct properties. These fundamental laws are not easy to comprehend, and therefore, extraordinary analytical abilities are required to understand their progression.
Induction and contradiction are basic operations of developing theorems and conjectures in number theory, and most of the elementary methods in the subject are constituted by the laws inherent in these operations. In general terms, induction is the process of generalizing any property of a mathematical entity (number, equation or function) by observing its operations over a specified range or value, while contradiction is the method of proving some operation (conjecture, hypothesis or simple assertion) wrong or right by simply proving its subsequent negation.
Groups and rings are two different algebraic structures that are derived from the operations of different sets of integers upon each other. More specifically, a group is any combination of integers or their sets in a single major set along with the arithmetic operation that makes them common to each other, while a ring is also a combination of more than one group associated with each other through arithmetic properties of addition or multiplication.
Prime numbers constitute the largest section in the study of elementary number theory, and their properties, functions and features cumulatively encompass some of the most basic as well as the most complex and difficult problems in the subject. Actually, prime numbers are those integers which are only divisible by themselves and 1 (like 2, 3, 5, 7, 11, etc.), and unlike other number sequences, they have no determined pattern associated with them. These features of prime numbers make them the most intriguing and mysterious (still unsolved) topic in all of number theory. At elementary stages, their introduction is necessary for all students to encourage them to do research.