The theory of numbers occupies a very unique position in the world of mathematics. This is one of the few disciplines which has a history beyond any academic house. The elementary number theory has been interesting for mathematicians and others because it can be formulated, modeled and varied to arouse the interest and curiosity of those having little or no mathematical background. By nature, number theory is a discipline that demands a high standard of training and so every aspect of this topic is presented with clear and detailed arguments.

The number theory in its elementary form with all the properties of the integers and more, would cover natural or positive integers, real numbers which will deal with operations indulging in plotting on a number line, presentation of whole numbers and the relevant applications in framing the fractions and their decimals, ordering of numbers that would help understand the elementary facts about natural numbers and uncovering the rules and forms of rational and irrational numbers.

Theorem 1 for Archimedean Property

If ‘x’ and ‘y’ are positive integers, then a positive integer would exist in such a way that nx = y.

Theorem 2 for Principle of Finite Induction

If ‘S’ is considered as a set of positive integers with the following properties then the integer ‘l’ would belong to S.
If there is an integer ‘k’ in S, then the next to this integer ‘k+1’ is considered in S as well.

Corollary to Theorem 2

If ‘a’ and ‘b’ are integers with b $\neq$ 0, then the unique pair of integers ‘m’ and ‘n’ are such that a = mb + n and 0 $\leq$ n < IbI
When, we have b < 0, $\frac{a}{b}$ = m + $\frac{n}{b}$ and 0 $\geq$ $\frac{n}{b}$ > -1

Theorem 3

If ‘c’ divides a1 …….ak, then ‘c’ divides (a1) (u1) + …. + (ak) (uk) for all integers u1,….., uk.
Also, aIb and bIa, if and only if a = ± b

Corollary of Theorem 3

If ‘c’ would divide ‘a’ and ‘b’, then ‘c’ divides ‘au + bv’ for all integers ‘u and ‘v’

Theorem 4

Let ‘a’ and ‘b’ be the integers with the greatest common divisor ‘d’, then an integer ‘c’ has the form of mx + by
(This is true for some of the ‘x’ where ‘y’ ? Z, if and only if ‘c’ is a multiple of divisor ‘d’.)

Corollary of Theorem 4

The two integers ‘m’ and ‘n’ are co-prime if and only ‘x’ and ‘y’ exists in “mx + ny = 1”