Euclid’s Division Lemma
For three positive integers a, b there exists a unique integer q and r such that a = bq + r and here value of r will always less then b
That means if we divide number a by b and q is our quotient and r is remainder then value of remainder will always less then deviser b .
For example 5 and 22
If we divide 22 by 5 we get 4 as quotient and 2 is remainder
So we can write it as 22= 5*4+ 2
And you can see that value of remainder is less then deviser
Every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.
The Fundamental Theorem of Arithmetic
If we ignore the order, any number which is more than 1, is either a prime number or can be written as unique product of prime number .
Every composite number can be written in term of product of unique set of prime numbers ignoring their
26 is a composite number and it can be written as 2*13 according to fundamental theorem you cannot get another set of value which product is 26.
So if we ignore the order of factors, prime factorization of all natural number is always unique.
Theorem of rational number: -
for any prime number p which is divides a^2 then it will divide a also.
For example: 5 divides 100 then it will divide root of 100( 10) also.
Theorem of terminator:-
for any rational number x which is written as p/q and if q can be written in form of 2^n*p^m and value of m and n are positive integer or equal to 0 then decimal expansion of x will terminate.
For example take x = 7/50
And 50 = 2*5*5 = 2^1*5^2
And powers 1 and 2 both are positive integer So value of x will terminate
Theorem of non-terminating repeating (recurring) :-
for any rational number x which is written as p/q and if q can’t be written in form of 2^n*p^m and value of m and n are positive integer or equal to 0 then decimal expansion of x will non-terminating repeating (recurring).
X = 7/15
Here denominator 15 = 3*5
And it cannot be written in form of 2^n*5^m
So x will non-terminating repeating (recurring) .