Question

Define Euclid's division algorith lemma and fundamental theorem of arthmetic

Answer 1

<br> Euclid's division lemma: For given any positive integers a and b there exist unique integers q and r satisfying a = bq + r, 0 ⤠r < b. Euclid's division algorithm is used for finding the Highest Common Factor of two numbers where in we apply the statement of Euclid's division lemma.  Fundamental Theorem of Arithmetic :- Fundamental Theorem of Arithmetic states that every composite number greater than 1 can be expressed or factorized as a unique product of prime numbers (ignoring the order of the prime factors). It is also known as 'Unique Factorization Theorem' or the 'Unique Prime-Factorization Method.  

Answer 2

Euclid's division algorithm :-


Given two positive integers a and b , there exist unique integers q and r satisfying ,

a = bq + r , 0< r < b.


Here,

a and b are called dividend and divisor respectively, q is called quotient and r the remainder.




• The Fundamental theorem of Arthmetic



1. Every composite number can be expressed as a product of primes and this factorisation is unique , apart from the order in which the prime factors occur.



2. . If a prime number p divides a² , then p also divides a where a is a positive integer.


3 . A rational number with terminating decimal expansion can always be expressed in the form of P/q , where p and q are coprime and the prime factorisation of q is of the form 2^m . 5^n where m and n are non-negative integers and vice versa.