Question

state the fundamental theorem of arithmetic and prove it.
(no spamming)

Answer 1

Theorem (Fundamental Arithmetic Theorem) : 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. 


1) Find the LCM and HCF of 6 and 20 by the prime factorization method. 
Solution : 
We have : 6 = 2 × 3    
 20 = 2 × 2 × 5 
20 = 2 2 × 5.
Common factors of 6 and 20 are 2 1 and 2 2 
So for HCF take the common number with lowest exponent.
∴ HCF = 2 1 = 2 
In LCM take the common factor with highest exponent and the remaining factors.
∴ LCM of 6, 20 = 2 2 × 3 × 5 
∴ LCM = 4 x 3 x 5 = 60

Answer 2

Fundamental Theorem of arithmetic states that
Every composite number can be expressed in the form of its product of its prime and the prime factorisation is unique