State unique factorization theorem and express is with example 196?
Answers
Answer:
196 = 2 × 2 × 7 × 7
Step-by-step explanation:
Statement of Theorem:
Let n be positive integer other than 1 then n can be express as the product of prime numbers which are unique up-to permutation
That is
n = p1 × p2 × p3 × ... × pr
where
p1 , p2 , p3 , ... , pr are prime numbers
Meaning of unique up-to permutation:
It means that if there are two representations for any positive integer n
then they must be same but may have different permutation means they are may be in different order
For example
Let n = 6 then
we have following two representation for 6
6 = 2 × 3
6 = 3 × 2
Note that prime number are unique but have different representation or permutation.
I hope you understand the meaning of unique up-to permutation
Example given in the question
To find the prime factorization for 196
196 = 2 × 98 = 2 × 2 × 49 = 2 × 2 × 7 × 7
So prime factorization of 196 is 2 × 2 × 7 × 7 that is
196 = 2 × 2 × 7 × 7