a Ring is called Gaussian Ring if: R is an Integral Domain. R is a Unique Factorization Domain (UFD), i.e. every non-zero non-unit element in R can be written as a product of irreducibles of R and The factorization into irreducibles is unique up to the order of the multiplication or the associates of the factors. Hope this has helped anyone Sagy

Unique factorization usually means that any integer can only be factored in one way using prime numbers only:

24 = 2 x 2 x 2 x 3 (unique prime factorization)

If other numbers than prime numbers are allowed, factorization is not unique.

24 = 2 x 12 = 3 x 8 = 4 x 6 = -4 x -6 = etc. (non-unique factorization)

If 1 is allowed, then every number has an infinity of factorizations:

5 = 1 x 5 = 1 x 1 x 5 = 1 x 1 x 1 x 5 = etc.

So, limiting the allowed factors to prime numbers, makes the factorization unique.

The theorem is that every integer has a unique prime factorization. So, the answer to your question could be any number showing its unique prime factorization.

No. Each composite number has its own unique prime factorization.

In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integergreater than 1 either is prime itself or is the product of prime numbers, and that this product is unique, up to the order of the factors.

Every positive composite number only has one unique prime factorization.