Elementary number theory is a theory of pure mathematics which is related to study of number systems (mainly Integers).

Every system of numbers has its limitations and this motivates to extend some more number systems by including some more numbers to it.

The limitation of natural numbers led to the discovery of integers and the limitation of integers led to the development of the set of rational numbers and so on.

__Some Important Definitions, Results and Theorems of Elementary Number Theory are as follows:__** Divisibility in Integers:**
If a and b are any two integers such that a not equal to zero, we say
that a divides b, denoted by $\frac{a}{b}$, if b = ac, for some integer c.

1. $\frac{c}{a}$ and $\frac{c}{b}$

2. If d is any integer such that $\frac{d}{a}$ and $\frac{d}{b}$, then $\frac{d}{c}$.

The g.c.d of two integers a and b is denoted by (a, b).

Two integers a and b are said to be relatively prime if the g.c.d of a and b is 1. i.e., gcd (a, b) = 1.

Its corollary is, let p be a prime number such that $\frac{p}{(a_1, a_2,… .., a_n) (a_i\ represents\ integers)}$. Then $\frac{p}{a}$, for some i, 1 $\leq$ i $\leq$ n.

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First we will find greatest common divisor of 243 and 189 by division algorithm, which can be written as 243 = 189 $\times$ 1 + 54 .......(i)

189 = 54 $\times$ 3 + 27 ……….(ii)

54 = 27 $\times$ 2.

Therefore, (243, 189) = 27.

From (ii) 27 = 189 – 54 $\times$ 3

= 189 – (243 - 189) $\times$ 3 by (i)

Therefore, (-3) $\times$ 243 + (4) $\times$ 189 = 27.

Hence x = -3, and y = 4 is the required solution.

Let d = (a + b, a - b), then $\frac{d}{(a+b)}$ and $\frac{d}{(a-b)}$ this implies $\frac{d}{(a - b + a - b)}$ and $\frac{d}{(a - b - a + b)}$. This implies $\frac{d}{2(a, b)}$ = 2 $\times$ 1 = 2.

Hence, $\frac{d}{2}$ implies d = 1 or 2.