Question

state and prove Unique Factorization Theorem

Answer 1

Unique factorisation theorem, also known as Euclid’s lemma, states that every real rational whole number can be expressed as the product of a group of prime numbers which form a unique set.

Let us assume a whole number S is written as follows:

S = p1 x p2 x p3 … x p(m)

S = q1 x q2 x q3 … x q(n)

To prove the UFT, it is sufficient to prove that m = n and that q1, q2, q3… q(n) are a rearrangement of p1, p2, p3… p(m).

If we divide S by p1, we must have a similar quantity in the set of q values. Since q is a prime number, its only factors are 1 and itself. Hence, by relabeling the values of p and q,

p1 = q1

Similarly,

p2 = q2

p3 = q3

…

p(m) = q(n)

This occurs only if m <= n. By reversing p and q, we derive that m >= n. Hence, m = n.

Hence, UFT has been proved.

Answer 2

The Unique factorisation theorum states that every whole number greater than one is a product of a unique list of prime number.Even the list of the factors

can have the same prime number in it more than once.According to this theorum even if you change the order of the list of the numbers even then it will

be consideed as same list

There is only one possible list for prime number factors for any original number.

No matter how you go about doing prime factorisation,you will endup getting

the same result.